Pseudo-Normed Linear Sets Over Valued Rings Citation LaSalle, Joseph Pierre (1941) Pseudo-Normed Linear Sets Over Valued Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7p59-bd80. https://resolver.caltech.edu/CaltechTHESIS:02242025-211405907 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:02242025-211405907 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02242025-211405907 DOI: 10.7907/7p59-bd80 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17020 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Feb 2025 00:18 Last Modified: 25 Feb 2025 00:24 Thesis Files PDF - Final Version See Usage Policy. 23MB Repository Staff Only: item control page

Pseudo-Normed Linear Sets over Valued Rings Thesis by J.P. LaSalle In partial fulfilment of the requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1941 Acknowledgement I wish to express my gratitude to Professor Michal for the advice, guidance and encouragement that he has at all times given me~ SUMMARY The main purpose of this thesis has been to investigate to what extent some of the theorems in the study of linear topological spaces are actually dependent on tbe real number multipliers. This has been done by replacing the real number system by a valued- ring, and, since such a ring may have a discrete topology, the results that we obtain depend only on the algebraic properties of our generalized number system .. In Chapter 2 there is developed a characterization of a topological space in a form which is convenient for the purposes of this paper. Chapters 1 and 3 are taken up with the introduction of the definition of the type of space with which we shall deal. In these chapters the author was guided by Hyert' definition of a pseudo-normed linear s pace 1 and Michal's definition of a topological :t ft-ri1Af. abelian A In Chapter 4 a generalized study of linear functions is considered. Here we obtain results which reduce to knovm results in the case of Banach spaces. Chapter 5 deals with the concept of a differential and a relationship between this differential and an M1 -differential2 is established. In Chapter 6 we strengthen our postulates somewhat and carry on the study of linear functions and differentials. In some respects we have accomplished the purpose of the thesis, though the author feels that much more can be done and hopes that this thesis may prove to be a basis for further work. • 2 See [lo] • CONTENTS BASIC DEFINITIONS ................................. page 1 CHAPTER 1. LINEAR SETS OVER VALUED RINGS. 1.1. Linear sets over rings with a unity element. V- spaces................ ... . .. . 1.. 2. Valued rings.. . ............................. V1 -spaces................................... I.e.. CHAPTER 2. NEIGHBORHOOD TOPOLOGIES. 2.1.. N-spa. ces .................................... 2.2. Equivalence of N-spaces and topological 2.3. N1 -spa. ces.a•••· e•• · • • • a • • · • • · · · · · · · · · · ·•••••• spaces"........... . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3. 3.l. 3.2. 3.3. 3.4. 3.5. J.7 19 26 SPACES PSEUDO-NORMED P-spaces. Equivalence of N-spaces .and P-spaces... . . . . . . . . . . . . . . . . . . . . . . . . . To polo gical properties of P-spaces.......... P-groups.................................... 33 P1 -~paces••················:···•····•······· Equivalence of P1 -spaces ana. N1 -spaces .. .... 36 38 CHAPTER 4. 4.1. 4. 2. ;7 10 15 28 31 cl -LINEAR FUNCTIONS ON Pl-SPACES TO P1-SPACES I -linear functions......................... Sequences of i - linear functions.. . . . . . . . . . 46 53 CHAPTER 5. J -DIFFEREN TIALS OF FUNCTI ONS WITH ARGuiiiliN TS AND VALUES IN P1 -SPACES 5.1. I -approximation functions.................. 58 5 .2. 5 .3. 's....... 62 68 ~ - differentiable functions................. J -d ifferentials in Michal ts t .a.. g. CHAPTER 6. PSEUDO-NORMED LINEAR SETS OVER VALUED DIVISION RINGS. P2- SPACES 6.1. 6.2 .. 6e3e P2 - Spa Ce S • • • ~ ••••• e • • • • • • • • • • • • • • • • • • 11 • • • • • • Linear function on Pn-spaces •............... I-differentials of function with ar guments and values in P -spaces ................ . 2 BIBLIOGRAf)HY • .......... c. • • • • • • • • • • • • • • • • • • • • • • • • • • 76 81 84 89 BASIC DEFINITIONS It is, of course, impos s ible to state here all the definitions to which we shall wish to refer; but, for the sake of being definite and as a matter of convenience to the reader, we shall list those definitions and theorems which are fundamental to this thesis and are not contained in the body of the paper. LI l and ll] are references in algebra and topology which are well suited to the needs of this paper. The concept of a set shall be as s mo.ed to be known, and we shall consider a set S to be defined when given any object pit can be determined whether pis or is not contained in S; i.e. either p e. S or p E S. We are also assuming that for each set there is defined a fundamental equivalence relation which we shall call equality and shall write as n= 11 • Two elements p and q of a set Swill be said to be distinct if p / q. All the sets that we shall consider will be assumed to contain at least two distinct elements. be. Our equality is also to/\such that, if p = q, then any Hstatement" that vve shall make a bout p can also be made about q. This means that though we do not require our equality to be identity, we do require that elements which are equal have certain properties in collilllon. These certain properties are 3.ny of those which happen to arise in our postulates, since these properties are the only oms which intere s t us. For instance, if in I(g ) below .we have x = y, then we can state that (-x) + y = a. We also are able to prove certain theorems about our equality, such as in I(h) and I(i). concept of a function The and an operation which we shall adopt will be the same as that in C1] pp . '.J- 4 , though it is to be remembered that uniqueness is now in terms of our equality. I Groups. A set G of elements x,y,z;••• is called a group with respect to an operation 11 +1:(written additively) if the following conditions are satisfied: a. x + yE G for every x,ye G; b. x + (y + z)= (x + y) + z for all x,y,zE.G; c. there exists an element 6 E G such that x + 8 = x for all X € G; d• that X 8 is called the ~ element of G·, given X€G t here exists an element -x c G such + (-x) = El. If, in addition, e. X + y = y + X for all x,y €: G, the group is said to be abelian. Then it follows from the above postulates that . f. El + X = x for each x ~ G; g. (-x) + X = e for each xE G·, h. the zero element e is unique, i.e. if 8 ' is any other element of G satisfying (c), then 6 = El '; i. the inverse -x of each x E G is unique; j. -(-x) = x. II Rings. A set A of elements a,~,,, ... is said to be a ring if A is an additive abelian group such that a second operation(written as multiplication) is defined such that: a. a ~ ~ A for all a,~ E A; b. ~(~'() = (af)'( for all a,~,Yc A; c. a ( ~ + ¥ ) = a ~ + a l and ( a +~ ) Y = a r + (3 r for all a,13, 1EA. A ring A is said to have a unity element if there exists an element 1 in A such that la =al= a for each aE A. 1 is then unique. It can be shovm that for any two elements a,(3 EA that: d. Oa= aO = o, where O is the zero element of A; e. (-a)~= a(-~) = -(a~); f. (-a)(-~)= a~. A division ring ( or quasi-field) is an additive abelian group whose non-zero elements form a multiplicative group such that II(c) is satisfied. III Integral domains. An integral domain A is a commutative ring (i.e. a~=~ a for each a.,~eA ) with a unity element and no divisors of zero(i.e. a~= 0 implies that a = 0 or~ = O. IV Fields. A field A is an additive abelian group whose non- zero elements form a multiplicative abelian group such that . a ( ~ + '( ) = a~ + a¥ for all a,~,¥€ A. V Linear sets™- rings. An additive abelian group G of elements x,y,z, ... is called a (left) linear set™~ ring A of elements a,~,1,••• if there exists an operation ax on AG to G such that: a. a(~ x) = (a~ )x ; b. (a + ~ )x = ax + ~ x ; c. a(x + y) =ax+ ay for all a,~ e A and x,y e G. If A has a unity element, we shall also require that lx = x for each x E: G. VI Strongly partially ordered sets.(Directed systems.) set D of elements d,e, ... A is said to be a strongly partially ordered set if there exists an order 11 / 11 defined between some of the elements of D such that: a. if e) d, then it is not true that d) e ; b. (transitivity) if e) d c. given e and d from D, there exists a cc D such that C >- e and C~ , and d) c , then e > C • d. A set D and an order n > 11 satisfying (a) and (b) is said to be partially ordered. N.B. It is not required that the order between each pair of elements of D. imply that e/d 11 > 11 exist does not necessarily d~ e. If E and D ~ two strongly partially ordered~,~ i t ~ easily~ proved .:thai ~ product space ED = [ ( e, d) ; e E E, d E nJ is illQ a strongly partially ordered set with (e 1 ,d 1 )~ (e ,d 2 ) 2 5 if and only if e 1 ~ e 2 and e1, e 2 f E . dl > = d2 for d ,d E D and 1 2 if and only if e 1 = e 2 and (e 1 ,d 1 ) = ~ ,d 2 ) 2 dl = d2• Hence, if Dis a strongly partially ordered set and R+ is the set of all positive real numbers with ordering as usual, then is a strongly partially ordered set with (ei = ri di E: E ) el ~ e2 if > and rl = r2 Define and di.= d2 • VII Topological s12aces. > dl = d2• ld = d el= e2 if rl = r2 and r 1 (r 2d) = (r 1 r ) d. 2 A set T of elements x,y,z, ... is said to be a topological space if: a. To each set M of elements of T t here correspond s a set M which is called the closure of M. b. For every xE T , x = x. c. If M and N are any two subsets of T, then MuN= :M uN. d. M = M for every subset Mc T. The elements of Tare called points. VIII Limit points. A point x of a to po logical space Tis called a limit point of a subset Mc: T if I X Open and closed sets. A set M of a to pological s pace is said to be closed if M= M . A set x6 MT x . Mc T is said to be open if the complement of M, CM = TT M , is closed. T 6 X Complete system of 11 neighborhoods 11 • sets of a topological s pace A system E of open T is called a complete system of Hneighborhoods 11 of T if every open set of T can be obtained as a sum of open sets belonging to containing a point E. Every open set of E x E:. T is said to be a 11 neighborhood II of x. N.B. This notion of a neighborhood is stronger t han we shall wish to use. XI See Chapter 2 . Limit of~ sequence of points. of elements of a to pological s pa ce Let fxn1 be a se quenc e T • The sequence { xn1 is said to have a limit if there exist s an element such tha t for ea ch open set a positive integer n> m. XII Uxc T , m = m(Ux) xnE Ux for all lim xn = x. Continuous functions (mappings) Q!l a topological space T to~ topological space T' . a topological space be continuous on T A function(mapping) to a topological space T to T' f (M)c f (M) XIII x E. Ux , t here exists such t hat We shall denote this by x E: T Topological groups. if f or every set T' f(x) on is said to lVi c T • A group G is said to be a tono- logical group if G is a topological space such that the group operations x + y and -x are continuous in G. CHAPTER l LINEAR SETS OVER VALUED RINGS. 1.1 VECTOR SPACES. Linear sets™ rings with~ unity element . Y- §paces. T over a ring 2 A with a DEFINITION l.ll. A linear setl unity element will be called a vector s2ace with respect to A, or briefly, a V-space w. r.-1_. A. We see i mmediately that a V-space is a generaliza tion of Banach's linear (vector) space 3 , where the real number multipliers have been re placed by multipliers taken from an abstract ring. We shall now pro ceed to investigate some of the elemen- tarJr properties of V-spaces. THEOREM 1.1.1 . If all X If and T T a is ~ V-snace w. £.• 1_. Ox = e , (1.12.) (-a)x = a(-x) = -(ax ) (l.J,3 .) ae = e • is ~ ( l.14.) , then for A ( 1.11.) A A , division ring4 , then ax= e i mQlies a= 0 Q£. X = e • l See V. Roman numerals refer to the sections of basic definitions. 2 See II. 3 See [ 31 ... Numbers in brackets refer to the bibliography at the end of the paper. 4 See II. 8 ax = (0 + a)x = Ox + ax, and since Proof: By V(b) group Ox = 8 ' and (l.11) has been proved. V(b) e = Ox= (a - o)x = ax+ (-a)x, and hence (-a)x = -(ax) • (1.13) follows from V(c) (l.12) in the following manner: Thereand a8 = a(x - x) =ax+ a(-x) = In order to prove (1.14) let us assume that A is a division ring and that exists an By (1.11) and = a(-lx) = (-a)x. V(a) and II(e) it follows that a.(-x) fore (l.12) has been proved. is a (-l)x = -(lx) = -x, and from It then follows from V(d) that ax - (ax)= e. T a-l E A such that ax= 8, a# 0. Then there a- 1 (ax) = (a- 1a)x = x = 8. ax= e, Hence, if we assume that x # e, we a# 0, and obtain a contradiction, and (1.4) is proved. DEFI NITION 1.12 . T , a5 , a E A and S, s1 , 82 , ... For -S , and as = are defined respectively by Sl + S2 [ ax; subsets of Xf S J, -S = -lS , and s + s = 1 THEOREM 1.12. S c 2 2 For all [x + y ,. x s1 , y s x , y '= T ; E E:: ] 2 all a , (3 6 A ; • and all T : ( 1.15.) s+e=s, (1 .16 .) sl + s2 = s2 + sl; ( 1.17. ) S + (sl + S2) = (S + S1) + s2 ~ (1.1s.) 18 = s, (-a)S = (a)(-s) = -( 0 s) , a(S 1 + s ) = as + aS ; 2 1 2 (1.110 .) a(? S) = (a(3 )S; ( 1.19.) ( 1.111.) (a + (' ) s = as + ~ s ; (1.112.) If (1.113. ) If conversely, if Proof: x - y € S, then + S , then XG y (1.15)-(1.17) and (1.13) 1.12 and the fact that x E y + S , and , x - y E S • follow directly from definition T is an additive abelian group. (1.18) - (l.112) follow from the fact that T is av~ space and definition 1.12. DEFINITION 1.13. a/~ s = [ Define for each x ; _~ x E N.B. as S c T , a , ~€ A , (3 "f O , J. It is clear that if A is a division ring, then -I a/~ S = a~ S • THEOREM 1.13. and For all a , \3 , 1 , a 1 E. A ; ~ -/: 0 , '{ -/: 0 ; s CT: (3 S c as = aS ; ( l.114 ~ a/l S ( 1 .115 .) al/(3 (as) = a. a/(? S ; \l.ll6.) a/~ (l.ll7 .) ~ (a,/~ ( l.ll8. ) if 1 implies that If ax= e S c. ( "{ a)/ ( '( ~ ) S ; s) c as Sc a/~ S, then~ Sc as, and conversely S c a/13 S . implies that (1.119 ~ a= O or ( '( a)/ ( Y~ ) S = a/(?, S . x = e, then Proof: (1.114) is cl early true. (1.115): y = az , xfa 1 /~ (as); Let z €: S . p x = a 1 (az) = ~la) z , and h ence Then a 1 /{3 (aS) c (a 1 a)/i3 S . f x = ~ 1 q)z , where i..e. Conversely, if z ~ S , and x e (a 1 a)/~ S , then t3 x = a 1 (az) . Then (a1 a)/~ S = a1/~ (a:S) • S , then f x ~ as , and by (1.112 ) (a1 a)/p S c a / tL (aS) , and hence 1 (l.116): x E a/r3 If )fx f '( aS . of theorem 1.12 Hence x €: (~ a)/(Yf) S , and 0/(3 8 C (ya)/('(~) S • (l.117): x ta/~ S , then (:3 x ~as , and hence If ~ (aliB s) c as • (1.118): This f ollows aln:ost by definit i on. (1.119) : If x e ( y o) / ( Yp ) S , then Y, (6 x = (' a:z , Hence '( ( ~ x - az) = 8 . and the a bove Since t :/ 0 , we have by hypothesis ( d' a)/ ( t <9 ) S c that ~ x = az , or z '- S . a/f-; S . By (1.116) aJf S = ( Ya)/ ( '((3 ) S . This completes the proof of theorem 1.13. 1.2. Valued rings. The next step in our generalizati on is to introduce the "number system" which is to re pl a ce the re a l number multipliers in the study_ of linear ·top9l6gical _s paces. Such a 11 number system!! will be called a va lued ring and is defined as follows. DEFINITION 1.21. If A is a ring with a unity elemen t such 11 that there is defined on A a real-valued function such that: /a/ then ( 1.21.) /a/~ 0 ll-22 .) ( l.23.) I a~ I ~ I a/ If I for all a,~~ A; I a + f3 I f I a/ + Ip I for all a ,~e-A ,. ( l.24 .) /-l/ = l ; ( 1.25 .) /a/> 1 for some for all at A ; is said to be a valued ring. A valuation of N.J1. a€: A , 1 d... ~ 0 /a/ is called the a. A shall Throughout the remainder of the paper a , (3 , ;r be used to denote g_ valued rigg_ with elements and with * the valuation function. /a/ ... Later in the paper shall put additional restrictions Q1l . A. THEOREM l.21. valuation function, then (1.27.) /l/ =- l , /-a/ = / a/ for all a t= A , l l.28.) /0/ f O implies that ( 1.26 .) > for all _ /a/= 1 aeA • Proof: I a/.= /la/ ~ /1/ / a/ for all fl €: A , and hence l f /1/. Also /1/ = /(-1)(-1)/ ~ /-l/ /-1/ = l . /1/ = 1. /-a/§ /-l/ /a/ =/a/. /-1/ / -a/= /-a/, and hence /0/ /a/ for all a e A , and if for all a tE: A • 1 Similarly, /a/= /-a/. Therefore /a/~ /0/ = /Oa/ ~ /0/ f 0 , then 1 &: /a/ I t i s a lways possib l e,without 8ny loss of general i t½ to def i ne / 0/ = O, and thr oughou t thi s paper we shall assume that /0/ has been so def i ned . 12 Examples of valued rings. (l) Valued fields. If A is a field which is a valued ring such that / a (3 / = / a/ I !3 / to be a valued f i eld. The concept of a valued field was then A is said introduced by Kiirschak 1 , and the conditions under which t he de r ived field of such a f ield be isomorphic with either the field of complex numbers or the field of real numbers are well known. Examples of such fields,and hence examples of valued rings are the field of real numbers, the field of cori.1plex numbers, and p-adic number fields. (2) example of a division ring that is a valued ring is the field of quaternions 2 , t hough here again we have An /af /=/a//~ I . (3) One of the more interesting instances of valued rings are t he and Martin in 11 linear normed rings 11 considered by Michal [1 ~J • There is given in t ha t paper an ex- ample of a linear normed ring of infinite dimension for / a(J / /: / a/ / f3 / which (4) for some elements. An example of an inte gral domain t ha t is also a valued ring is, of course, the integral domain of all inte gers, the valuation function being t he absolute value. (5) Let us now construct an example of a valued ring which is such that l 2 / a ;9 / =/=; / a/ I13 / See [ 7] , and also See [ IJ.-(], p. 172. for all a , f G A • Cha pters XI and XII 13 Let Q with elements p, q, s, ... be a ring with a unity element, and let a = (po' Pi, • • • ,Pn, • • • ) be a sequence of elements of sequences of elements of Q. Let A be the set of all Q which contain only a fin ite nurrober of non-zero elements of I.e., if Q • a r: A , f €, A then a= (p o , P1, . • . , p n , 0, 0, . • • ) and (qo, ql, ••• ,qn,o,o, ••• ) Define . = (p + q 'P1 + ql, .•• ) 0 0 and af = (s 0 , s1 , ... , sn + m,o,o, ... ) si =jTu=i (pjqk) • a= (3 if and only if pk= qk where for k = o, 1, 2, ... • Then it may be easily verified that A is a ring with a unity element where o = (o, o, ... ) and 1 = (1, o, o, ... ). -a = (-p 0 , -pl, If afA , a 1- 0 , then a last non-zero element of the sequence exists. Cle ar ly /a/, .. . ) . /a/ = 0 for alJ. pn Define /a/ = n + 1 , and a l A , /-1/ = 1 , /a +(3 / = max ( If/ ) , and there exist a€:: A such that /0/ = o. /a/) 1. If / a/ = n + 1 > 0 , / p/ = m + > 1 0 , then / af / = /a// ~ /a// f / , and /ap / = /a/ / /3 / implies either /a/=~~/= 1 , or /a/ = 0 or 1(3 / = 0. n + m + 1. Hence Hence~§.§.§. that the set A of all formal polynornials 1 Q with~ unity element is with coefficients in~ ring a valued ring, with the valuation@ defined above. (6) It is also possible to value t he field real numbers in such a manner that /r/ R of is zero or an integer, and also such that t he re exist r 1 , . r 2 f R /r 1 r 2 / ~ number /r1/ /r 2/ . p ) Moreover, there will exist no real such that 0 is the absolute value of Given r €. R ~ m + 1. m< r /r / = p I r I and there exist for all there exists an integer Defin e r cR r eR • \r I r . /r/ = m + 1. /r 1 + r 2/ ·J /r 1 / + /r 2/ , /-1/ = 1 , with m such that /r/ ~ 0 , Hence /r1r 2/ ~ /r1/ /r 2 / , with t he property that /r / > 1 /(1/2)(3/2)/ = 1 , whereas /(1/2)/ /(3/2)/ = 2, and hence R is valued as stated above. (7) the ring It should be noted that we have not assumed that A shall have no divisors of zero, and a simple example of a valued ring with divisors of zero is the ring of integers mod 4. The ring of integers mod 2 is, however, not a valued ring due to postulates (1. 24) and (1.25). 1 See [ 1] pp. 17, 18. 15 These examples of valued rings should indicate the extreme generality of our number system A, and examples (4) and (7) show, in particular, that A is not necessarily a topological space with a non-discrete topology generated by the valuation. 1.3 v1 - spaces. DEFINITION l.31. A linear set T over a valued ring A will be said to be a vector spac~ o f ~ ™ with respect to A or briefly a Vi-space ~-~•1• A. A v -space w.r.t. A is then a special case of a 1 V-space, and the set operations defined for V-spaces in definition 1.12 will be applied also to Vi-spaces and will, of course, have the same properties as before. N.B. It is evident that every additive abelian group is a Vi-space w.r.t. the i ntegral domain of all integers, where the valuation is the absolute value. CHAPTER 2 NEI GHBORHOOD TOPOLOGI ES. N-SPACES. From now on the spaces that will be considered in this 1 thesis will be to pological spaces. We shall not begin by . postulating tha t t hese spaces are topological, and the aim of this chapter is to develop a new characterization of topological spaces by means of which we can more ea sily verify that our spaces are topological. This new charac- terization will also enable us later on to express certain of the topological cone ept·s i n a form which will prove to be conveni ent in what is to follow. The equivalent s r.o.c. e definition of a topological ~t ha t we shall give will be stated in terms of the properties of a s pecial system of subsets of a set, these subsets will be called nei ghborhoods, though we shall not require tha t t hese neighborhoods be open. sets, as is done by Frechet, 2 Hausdorff, 2 and Pontr-jagin. 3 However, our neighborhood systems will be equivalent to a complete neighborhood system4 of the topological space. Then in chapter 3 we shall obtain from this characterization a second one based upon a generaliza tion of Hyers' pseudo-norm. 5 In this manner we show that a generalization of t he idea of a norm can be made a basis 1 See VII. 2 See [ 15"] 3 See llyJ 4 See x. 5 See l (, 1 , pp. 33-39. p. 30, theorem 3 . , l~J , or [ q] • l'7 for the study of topological spaces. 2.1. N-spaces. DEFINITION 2.ll. and let L Let T be a set of elements be a system of subsets of x, y, z, •~•, T, the subsets shall be called neighborhoods, such that: (2.11.) There is associated with each element x E T a non-null subset neighborhoods of x , of Z Lx of subsets Ux , called such that for each Ux E Zx XE UX • ( 2 .12 .) (2.13.) VxE i:x WYf. Zy that y # x , there exists a Ux E Zx y E. Ux • such that a If If UxE Zx and such that WX C Given x E T and such that if such that UX n VX • y <:: Vx WY c Ux . UX E ZX , there exists there exists a (Note that this implies Vx C Ux • ) The set T will then be called a space with a neighborhood system or simply an N-space w.r.t. Z. I Postulates (2.11) - (2.13) are those of Frechet, wher eas, postulate (2.14) is weaker than Frichetts fourth postulate. exists a Frechet requires that, if y E ux, then there U £ L y y such that Uy c Ux. Pontrajagin's conditions are even stronger, as he considers a system L such that if of x. and U E- I x G:: U , thet'I U is a neighborhood Hence (2.14) is automatically satisfied. An example of such a neighborhood system in which L the neighborhoods are not necessarily open sets would evidently be furnished by t aking . as neighborhoods in a Banach space 1 the closednspheres 11 generated by the norm. THEOREM 2.11. T is~ group (written additively) If such that there exists a system of subsets 1). = f U f , Uc T, satisfying the following postulates: 2 ( 2.15.) the intersection of all ( 2.16.) g_iven WE ll g_ such ( 2.17.) y E. V then is an N-space with and L Proof: there exists a U EU that if T there exists a = [ Lx ;. x E T V tlt there exists Wc U n V ; that given and U~ 11 U E 'U is e ; WE 11 Lx J. such that VE. it such y + Wc U ; = [ x + U ; U E u.J , In order to prove this we shall show that r as defined above satisfies (2.11) - (2.14). (2 .11): x E x + U for each (2.12): U EU Since for every If such that U E 11 U f. 1/. , e 6 U , we have · that • x ~ y, then by (2.15) there exists a -x + y ~ U • Then Ye- x + U and is satisfied by L. 1 See ( 31 , p. 53. 2 We shall call such a group an N-grou2 ~-~-t <"\.A.. (2 .12) 19 (2.13): This follows i mmediately from (2.16). (2 .14): Given U <=: ll and that there exists a Ve U such that if y E. x + V, there exists a or y + Wc x + U. W €U. x e T , we have by (2 .17) -x + y6 V , i.e., such that -x + y + W CU Hence (2.14) is satisfied by Z, and T is an N-space. Equivalence of N-spaces and topological spaces. 2.2. DEFINITION 2.21. w.r.t If S is any subset of an N-space Z, the closure of S, denoted by T S, is defined as follows: x € S if and only if for each . UxE Zx , s () ux ~ oo • 1 LEMMA 2.21. Every N-space T w.~.t . Z is~ topological space with closure defined as in definition 2.21. Proof: Our proof consists in showing that this operation of closure satisfies VII(a) - VII(d). (VII(a)): Follows by the definition of S . (VII(b)): Follows from (2.12) of definition 2.11. (VII(c)): Let T , and assume that ux intersects or S2 • Hence s1 and 82 x c S1 U S2 • S1 V s2 Then, for each Uxc Zx , , since each ux intersects Sl xE. s1 u s2 1 Throughout this paper be any two subsets of and 11 0°n s1 \J 82 c. s1 V s 2 • will denote the null set. To prove the converse assume that given Ux n (s1 u s2) :/- 0° ; UxE Lx , Ux() s 2 :/- 0° , or both or X 6' s1 u 82 . Lx Ux() 8 1 :/- 0° and subset { vx) C Vx f' S1 1 0° and Vx 6 Ux{I s -:/- 0° • 2 of subsets Ux e:. { Ux J such that for every uxn s2 :f. 0° • and Uxn s1 -:/- 0° i .. e., either Now assume that there exists a subset {ux] contained in Then, Ux {I 81 = 0 ° Similarly assume the existence of a such that for every LX Vx n s2 = oo • Given vx t: { vx} UXE 1ux} and 1 vx} , we have by (2.13) of definition 2.11 that there exists a wX E: zX intersects s1 such that or wX either Now wX cuxnv. X s2 , or intersects both s1 s2 . and This contradicts our assumption of the existence of 1Ux ~ , since wX is contained in both ux and lvx} vx • Hence every ux E. zx ei ther intersects Sl or inter- and S2, or intersects both sects XE Sl and s 2 • Then s1 u 82 c -s1 u -S2 • Since we also s1 u S2 C s1 u S2 , s1 u s2 = -s1 u -S2 , 81 U 82, and therefore have shown that and VII(c) is satisfied by our closure. (VII (d)): From our definition of closure and (2 .11) of definition 2.11, we have that Hence S c. S c S . sects S. If Sc S for each Sc T. x E S , then every Ux E ZX interNow by definition 2.11, postulate (2.14), there exists, corresponding to each uX CL X , a VX e L X such , there exists a Wy E zy such that wy ux • 8 , V Since we are assuming that also intersects that, if y E. vx C XE X S, and, for Then Ux intersects S = S. N.B. intersects S xe S . S Hence WY c Ux. and Sc S and Tnis completes the proof of the theorem. Throughout the remainder of the paper when we speak of to pological concepts in an N-space that T T, we shall mean is a topological space, closure being that of definition 2.21.. THEOREM 2.21. A necessary and sufficient condition that a subset S for each xe S Proof: of@ N-spac e T Y!•£.-t. t here exists a Sufficiency: If Ux E. ~x L be open is that Ux c S . such that x E S , we have by our sufficiency Ux c S, hypothesis tha t there exist a Uxe Lx and hence by definition 2.21 x ~ CS , which is equivalent Sn CS= 0° . to saying that follows that CS= CS. closed and Sis open. Necessity: Then, if xE S , not exist a for each Since such that CS c. CS, it then Therefore by definition CS is 1 Let S be an open set, i.e., CS x E CS = CS . Ux e Ex such that is closed. Suppose that there does Ux c S • Ux G Lx, and by definition 2.21 Then Ux " CS f. 0° xECS. This is a contradiction, and t he necessity is proved. DEFINITION 2.22 w.r.t. 1 2 L, 2 , If T is an N-space w.r.t. Land also then the two neighborhood systems ~ and L' Se e IX. More gener ally, this definition shall apply if ~ 1 sat i sfy (2 .11). r and will be said to be eguiy~lent, if for each x E T - •Lx and L¾ have the property that given Ux € Lx there correspond Ux' e Lx' such that Ux' c Ux , and conversely. li. B. It may easiiy be verified that, if w.r.t L E E' and and also w.r.t. implies that is an N-space T L 1 , then the equivalence of E r• and give rise to the same operation of closure. THEOREM 2.22. T is a11 IT-spacg_ ~.;r_ • .t,. If l is equiva.ient tq_ g_ complete system of r of nej_ghborhoods T. Proof : It follow s from (2.14) that given such tha t y E CUx given Wye Ux. Hence open set. x r' t: , then Then, if Uxe: rx, ~ rX there y tE Vx , we see that CUx c CUx , then C(CUx) c: Ux . Therefore, there corresponds an open set containing and contained in by theorem 2.21. Ux. The converse follows i mmediately The set of all open sets is a com~lete neighborhood system of T, hood of a point X Vx c:: C(CUx) , which is an y E C(CUx), and Since U x if s an open set Abeing a neighbor- x E. S , and the theorem is proved. COROLLARY 2.21. 2'. and at the same time an N-§J)ace w. r.t. r• , then§:. necessary r• give rise to t h e ~ and sufficient that L and 1 See X. It is to be remembered t hat in this case the sets U e i:' are open and t hat if x £ U then U is said to be a neighborhood of x ; i.e. Ex' = [ U ,; U e I:' , x E: U J• operation of closure is that Proof: L and L1 be equivalent. The sufficiency has been mentioned before, and the necessity follows from theorem 2.22 upon noting that the equivalence of neighborhood systems is an ttequivalen ce relation 111 and that complete neighborhood systems are equivalent. As is/shovm in [,11-(1 , p. 29, section D , every complete system .L of neighborhoods of a topological space satisfies (2 .1.2) and (2 .13) . An open set a neighborhood of a point x if U is now to be considered x E U, and therefore (2.11) and (2.14) are also satisfied by an N-s:pace w.r.t. L. T is then L, and we may summarize our results in the following theorem. THEOREM 2.23. Every N-sp ace T ~.r.t. L is~ to2ological space with closure defined as in definition 2.12, and L is equivalent to thg_ set of all opep sets in T. versely, every topological space _T a.ny system of N.B. is an N-spa ce 1!.;t.t. L equivalent to~ complete neighborhood system T. It might be pointed out here that if N-space w.r.t. then L and if each UE L T is an is an open set, L is a complete system of neighborhoods of This follows immediately from theorem 3, p. 30, in 1 Con- T. (IJ{ ] • See [ 1) , p.5. Here we make use of t he t r ansitivity of the equivalence rel a tion. 24 THEOREM 2.24. a function A necessar~ and sufficient condition that f(x) Q!1@ L' N-space T L j;;o an;y_ N- Yi• r_. :t_. be continuous 1 ll tha~ fQI:.. ~ wace Tt Yi.•!:.• t. xcT and Uf (x) E Lf (x) ' there correspond UXE LX f(Ux) C uf(x) ' • This theorem follows immediately from theorem such-that Proof: 2.23 above and theorem 4 of DEFINITION 2.23. A function x E Sx , of an N-space Z1 y G Sx if given u;(y) E ~t(y) L on an open set Sx, to any N-space T' will be said to be continuous at§:. noint Uy C Sx ' such that there exist Uy E Ly , f(Uy) C Uf(y). It then follows from theorem 2.24 that, if is a function on an N-space f(x) f(x) w.r.t. T w.r .. t. N.B. , p. 34. is continuous on T to an N-space T to is continuous at each point T' f(x) T' , if and only if then f(x) x € T . We saw in theorem 2.1.1 that every N-group is an Nspace, and now we wish to consider the case in which a group T is an N-space w.r.t. an N-group w.r.t. }.:e. Z and at the same time The question then arises as to whether the two neighborhood systems, [ x + U9 ; l See XII. x ET , Ue E ~~ J, :E and z, = are equiva].ent or., what is the same thing, whether the operations of closure that are defined in terms of the two neighborhood systems are the same. By theorem 2.11 and definition 2.23 we may state the following corollary. COROLLARY 2.21. t and if is also an :N-grolill, Yi•!:.• i• T necessary and sufficient condition that t' - [ X and E be equivalent is that: xET given and If t , X E T , ue ~e J ' where L~ = [x + Ue ; + Ue • UeE ~eJ, y = e re , illfill §;_ -x + y , X +y is continuous in is continuous ~n y at y at y = x. T is a topological group, the conditions of corollary 2.21 are satisfied, and hence in this case we are justified in always taking our neighborhood system to be generated by the neighborhoods of the origin in the manner above. THEOREM 2.25. A necessary and sufficient condition that 1 lim xn = x , where xn is a seauence of elements of j an N-s~ace T ~.r.1. exists @ integer that I L, is that given UX EtX m = m(Ux) such that there n) m implies xn E: Ux • Proof: 2.23. l See XI. The theorem is an immediate consequence of theorem 26 We shall now introduce our generaliza tion of a linear topological space. As in the deYelopment of the study of linear topological spaces, 1 we shall first define the space in terms of a neighborhood system and later consider an equivalent chara cterization in terms of the existence of a special real-valued function, which we shall call a pseudo-norm. DEFINITIO N 2.31. Let T be a v1 -space 2 with respect to A such that there exists a system U of subsets Uc T satisfying the following postulates: ( 2 . 31.) the interse ction of all ( 2.32.) given a WE-U. such that V ~ 11 ,. u £ 1(_ there exists such that pV Cu for all given u E- 11. there exists a /p/ f. /a/ ; VE ll such , V+Vc u . given such· that XET r es:gec t to A, u and u 611 an N-s pac ~ of ~ there exists an x Eau • We shall call such a space 2 t here exists and ( 2.35. ) a.EA W c U AV e ; VE ll a EA ( 2.34 . ) that and given ( 2.33 . ) a u ~ 11 U E 11 is T or simply an N.1 - s pac e Y!.•!:.•i• See definition 1.31. 1 with A, 'U • 27 THEOREM 2.31. N-group ~-!:.•1• 11. . 1 abelian gJ'.Q1H2.• Proof: T A, U is fill and is, in particular,~ topological It follows from (2.31), (2.32), and (2.34) that is an N-group w.r.t. U. . 2. 23 T ~-r.-1 Every N1.-space T By theorem is a topological space. abelian group. By 2.11 and theorem T is by definition an theorem 2.11 and theorem 2 . 24 we have that (2.33) and (2 .34) imply the continuity of t he operations x + y we take and -x. a= p = -1. In using (2.33) in the above Hence T is a topological abelian group, and the theorem is proved. · A further study of this type of s pace will be carried out in the next chapters when we consider the equivalent characterization of N1 -spaces. CHAPTER 3 PSEUDO-NORMED SPACES. P- SPACES. Hyers 1 has shown that the notion of a. pseudo-norm, i. e., a non-negative real-valued function defined for each element of a space and each element of a. strongly partially ordered set, 2 is sufficient to characterize linear topological spaces. In this chapter we wish to go further and show that a generalization of the form of Hyers' pseudonorm is fundamental to the study of topological spaces. The pseudo-norm in some cases may seem to be trivial, since it may be only another notation for set inclusion, though the value of our characterization lies in the fact that it takes as the basic concept in topology the extension of a notion which is a very familiar one, namely, that of a norm. This new postulational basis for topological spaces is also in a form which is very convenient for the purposes of this paper, and some of the topological properties imply interesting conditions on the pseudo-norm, as has already been demonstrated in the case of linear topological spaces. 3 3.1. P-spaces . DEFINITI ON 3.11, l 2 3 Equival ence of N-spaces and E-spaces . Let See • See VI. See for instance T and D be sets with elements , , or • x, y, z, ••• and d, e, c, ... respectively. Moreover, D will be assumed to be partiall y ordered. 1 The set T will be said to be a pseudo-normed spac~ with D, or briefly a P-space w.r_.t. respect to exists a non-negative real-valued function TD D, if there /Ix, di/ on . such that • ( 3.11.) null subset there correspond to each of elements of DX (3.12.) D , a.nd //x,d.j/ ~ 1 1 DX with the same for each //y ,dX.I/ ~ 1 for all d X E; DX given and 1 a non- D such that is a strongly partially ordered set ordering as X E. T dxE Dx ; implies t hat y = X ; an ex~ Dx x E T d X E DX such that for each that //y ,e.jl ~ 1 y ther e exists with t he property there corresponds a dye Dy such d y ~ dX ,• that {3.14. ) for all d ·~ e implies that /Ix, el/ ~ /Ix, d// x E: T • /Ix, d // will be called the I2_seudo-nonm of x with respect to d. If DEFINITION 3.12. T is a P-space w.r.t. shall define U(d) = [ x; x E T , /Ix; di/ § 1 ~ = [ U (d) ; 1. See VI. d.c D J, D, we J, 6U and Lx = LU(dx) ; dx Dx J • E L will be called the neighborhood system generated gy_ the pseudo-norm. If Tis also a P-space w.r.t. D1 , then the pseudo-norms /Ix, di/ and //x,d 1/ / ' will be said to 1 be equivalent if they generate equiva lent nei ghborhood systems. N.B. A sufficient condition for the equivalence of t wo pseudo-norms is that given 1 dx e Dx' dxE Dx r such that //y ,d.j/ =Lo //y ,d;(/' and conversely. there correspond for all y E- T , Later we shall show that in a special case this condition is both necessary and sufficient. THEOREM 3.11. Y!•£.•i• Every P-snace L, where m:: the pseudo-norm. Proof: Lx Let L T ~-£.•i• D is fill N-space is t he neighborhood system gene-rated Conversely, every N-space ~-~•i• T be a P-space w.r.t. be as in definition 3.12. D, and let 6 6 and Then (2 .ll)and (2.12) of definition 2.11 follow respectively fro m (3.11) and (3.12) of definition 3.11. (3.11), (3.14) and VI(c) imply (2.13). (2.14) is a direct consequence of (3.13) and (3.14). Hence the first part of the theorem has been proved. let 1 T be an N-space w.r.t. See definition 2.22. L Now , and we shall show that 6 .l (3.11) - (3.14) follow from (2~11) - (2.14) . Fir s t, it is evident that we can define an order for fying VI(a) and VI(b) by taking U >V if US V if t U c V, and U, V E t . U is properly contained in V , where x ET Then for each U E 2:: and 1/x,lW'~ l, if satis- define , xcU and /Ix, U// ~ 2 ~x , if x E U • is a strongly partially ordered space by virtue of (2.13) , and therefore (3.11) is satisfied. (3.12) follows from the definition of //x, U// (2.14) evi- dently implies (3.13). and (2.12). (3.14) is clearly satisfied, and the theorem has been proved. 3.2. Topological proper ti es of P- s pace s. follows It then/from the above theor em and t heorem 2 . 23 that T w.r.t every P-space D is a topolo gical space, and conversely. COROLLARY 3.21. Every P-space T w.r.t. logical space with closure defined as follows: if S is any subset of T , then closure of S, if and only if for each exist yes x t:: S , the d X E: such that //y,d.j/~l, :!:_.e., D X t h ere ycU(dx). .Also L, the neighborhood system generated by the pseudo- norm, is equivalent to the set of all open sets of Conversely, every topological space T. T can be pseudo- normed w.~•i• ~ complete neighborhood system of T or w.~.t. any system of subsets equivalent tog_ comnlete neighborhood system. Proof: This corollary is an i mmediate result of theorem 3.11 and theorem 2.23. COROLLARY 3.22. g_ subset S to each xE S //y,d/f L 1 Proof: A necessary and sufficient condition that of g_ E-space T w.r.t. there corresponds a imnlies D be open is that dx e Dx such that yE S, ;i_.§_. U(dx) c S. That this follows is clear from theorem 3.11 and theorem 2.21. COROLLARY 3.23. A necessary and sufficient condition that~ function f(x) be continuous at x = y there exists a y ES is that given y , of ' E Dr(y) ' df(y) ' such tnat //f(x),dr(y) // '£. = 1 for all //x,d./1 Proof: This result is a consequence of theorem 3.11 and definition 2.23. COROLLARY 3.24. £. necessary and sufficient condition 33 that@ element x of a P-suace limit of a sequence given dxE Dx that fO.£. all Proof: {xn'\ T ~.I'._.t. of elements of T is that m = m(dx) such there exists an integer n) m D be the //xn, dx// f: l . The proof of the corollary depends upon theorem 3.11 and theorem 2.25. 1 P- groups. 3. 3. The pseudo-normed spaces that we will be dealing with later on will be groups, and we shall now introduce the concept of P-group which shall correspond to. an N-group in the same manner that a P-space corresponds to an Nspace. DEFINITION 3.31. Let T be an additive group with elements y z , ... ,. and let D be a stron ;.,·ly artially ordered ,., x 0 set of elements T d , e , c , p· . . . . We shall say that is a pseudo-normed group with re.spect to a P-group w.r.t. D , or simply D, if there exists a non-negative, real-valued function defined on TD satisfying the following postulates: for all ( 3.31 .) //8,d// f: 1 ( 3.32.) /Ix , 41/ ~ 1 for all d 6 D implies x = e , where 8 ( 3.33.) given de D ; is the zero element of d ED there exists an , T • e (;- D - - - -· -----------~- --· 1 For the rela tion between P-grour s and topolo gic al a belian gr oups see section 4.2. 34 such that if //y, di/ f: l , then there exists a c = c(y) E- D with the property that //y + x, el/ ~ 1 for all /Ix, c // !:: l ; e ~ d (3.34. ) for all x E: T • THEOREM 3 .31. and Every P-group yy_.r.t. U 1 where yy_.:r_. 1,. implies that /Ix, di/ ~ /Ix, el/ = [ x ; x t T , //x, d // ~ l ] U (d) V, = [ U(d) ; d E DJ m Proof : D is an N-group . Conversely, eve r y [-groun be nseudo-normed Y[_.r .1,. 71 . The verification of this theorem is quite similar to tha t of theorem 3.ll, and t he pr oof will be omitted. LEMMA 3. 31. x ET and T !&_i d e- D be§!. !:-group YY.•!:.•1.• defin e (We shall always write Ux (d) = Ux(d) • Proof : Let y €. x + U(d) • //-x + y, d // f: l • = [ y ; y e T , //-x + y, di/ !: 1 u8 (d) simply as x + U(d) Hence D, and given Then U(d) . ) -x + y l:: U(d) ye Ux(d) and y e Ux(d) , then //-x + y, di/ f: 1 -x + y E U(d) • Therefore Ux (d) c x + U(d) and x + U(d) c. Ux(d) • Conversely, if y E. x + U.(d) • Th.fill and Hence and the lemma is proved. From now on in this thesis we shall be dealing with P-groups, and hence it will be useful to surnmarize in the following corollary t he characteri zation of certain to po- J. logical notions in terms of the pseudo-norm of a P-group. COROLLARY 3.31. (a) Every P-group w.~.i- T D is topological space with closure defined as follows: §:. if S closure of is any subset of S , if and on:L;y: if for each (b) T. A necessary and sufficient condition that a S c T subset d ED be open is that given such that //-x + y, di/ what is equivalent Ux (d) c S • function f (x) P-group Tt given d'E D' 911 fill open set Y!.•~•i• Dt SY c T , y 6 SY , to a be continuous at df:D x =y is that with the property implies that // -f(y) + f(x),d'// ' ~ 1 . A necessary and sufficient condition that an (d) element x f: T be the limi t of a sequence is that given such that Proof: there exists = 1 implies ye S Q.!:. there corresponds a that //-y + x , di/= 1 T x c S A necessary and sufficient condition that a (c) of there = [ Ux (d) ; x ~ T , d {=DJ 1. is equivalent to the L set of all onen sets of §:. dE D such that //-x + y, di/ ~ 1 • exists _g_ y € S Also T , then x f S , the ) xn l of elements d E. D there exists fill integer //-x + xn, di/ ~ 1 for all n m = m(d) >m . This corollary follows from theorem 3.31, lemma 3.31, theorem 2.11, theorem 2 . 21, theorem 2.23, definition 2.23, and theorem 2.25. 1 This neighborhood system will be said to be the ne i ghborhood system generateci QY. the pseudo-nor_[_ . This is 3na lo gous to definition 3.12. As an example of a P-group, consider any valued ring A for which the va.luation satisfies the additional property /a/= 0 t hat normed w.r.t. implies that a= 0. Then A can be pseudo- the set of all positive real numbers, R+. This can evidently be done by defining /I a, r // = r / a/ for each a € A and r e R+ . As we have mentioned before, it is interes t ing to note that we can find examples of valued rings in which the topology generated by the va.luation is trivial in t hat it i s discrete; whereas, the valuat ion itself, from a more general point of view, is by no means For instance see examples (4) and (5) of § 1.2. trivial. 3.4. P1 -space s. In this section of the paper we will define vvhat we shall call a Pi -space and shall show the equi va.lence of 1 P1 -spaces and N1 -spaces . P1 -spaces will be related to N1 -spaces in much the same manner that pseudo-normed linear spaces 2 , are related to linear topological spaces. DEFINITION 3.41. A v1 -space 3 T w.r.t. A will be said to be a pseudo- normed £Q~ o f ~ l wi th r es pect to or briefly a ~1 - s pace ~-~•!• 1 ~. . See § ') 32 See [.~1 • 3 See definition 1.31. 4 See definiti on 3.31. A, D .f J. T • a is A,D, P-group 4 w.r. t /Ix, di/ , satisfies such that the pseudo-norm, D the following additional postulates: l 3.41.) xc T ; and a c- A /lax, di/ f:: /a/ /Ix, di/ for all d f D , ( 3.42. ) given that /Ix, el/ ~ 1 /Ix + y, di/ !:::: 1 ; ( 3.43.) and //y, el/ ~ 1 e f: D such implies that l given corres ponds an there exists an d ED r) 0 , aE A xE: T, and a y <E T and df D such that there x = ay, 2 //y, d// f:: 1 , and /a/ - /Ix, di/ ~ r ,• ( 3.44.J e E: D an a.EA and d~ D such that /Ix, el/ S 1 1/px, di/ f: 1 N.B. giv:en for all there exi st s implies that /p/ f=: /a/ . The postulates which are necessary to determine the properties of the pseudo-norm are then (3.32), (3.34), and (3.41)-(3.44), since (3.41) implies (3.31) and (3.42) implies (3.33). THEOREM 3.41. \ 3.45 ) X ~ T 1/-x, di/ = /Ix, di/ for all dE D and ; \ 3.46 ) /16, di/ = 0 for all dE: D ; 1 This :postulate is a stronger form of (3.33). 2 This postulate then makes it unnecessary to assume that lx = x for each x f; T • Given x f= T , x = ay for some afc.A and yE. T. Hence by V(a) lx = l(ay) = ay = x. 38 ( 3 .47 } e f D given a €. A and there exist and //y, el/ ~ l such that /Ix, el/ f:: 1 that //p 1x + p y, di/ ~ 1 2 deD for all /p / ~ /a/ 1 implies and /p2/ f:: /a/• Proof: By (3.45): (3.41), (1.12) and (1.24) /1-x, di/ ~ /-1/ /Ix, di/ = /Ix, di/ . Similarly, /Ix, di/ f: /-1/ /1-x, di/ = /1-x, di/ . Hence /1-x, di/ = /Ix, di/ • (3.46): This follows from (3.41) and the fact that 1 we are assuming that /0/ = O . By (3.42) we have that given (3.47): exists an e 1 €. D such that //x,e 1 // ~ 1 and //y ,e ;f f:: 1 1 Now by (3.43) we have that implies that /Ix + y, d // f: 1 . given e 1 E D and //x,e// f:: 1 Hence //p 1x,ef/ ~ 1 • and /p 2/ f:: a e; A there exists an implies that //px,e1 //f:: 1 /Ix, e // f:: 1 dE D there and //y, e // ~ 1 and //p 2y,e-jl f: 1 e ED such that for all /p/ f: /a/. implies that for all /p 1 / ~ /a/ /a/, which in turn implies that // p1 x + p2 y, di/ f:. 1 • This completes t he proof of theorem 3.41. 3.5. The equivalence of f 1 -spaces and N1 -spaces. THEOREM 3.5J.. 2. Every P1 -space_ 1 See footnote (1), p. I l • 2 Compare with theorem 1 of T [6] Y[.r_.t. • A,D is fill 39 N1 -space w•!:.• t. A, U , where U is defined as in theorem 3.31, (i.~., 7i is defined to be the neighborhood system Conversely every N. -space 1 be pseudo-normed with respect to g_ generated QZ the pseudo-norm.). w•!:.•i• A, U m strongly gartially ordered set D in such a manner that the neighborhood system generated~ ,the pseu.do-llQ.ffil is equivalent to 7l The proof of the first part of the theorem is Proof: almost immediate. ?J. and define U (d) . Let T be a P -space w.r.t. 1 = [ U(d); d f DJ , where A, D = [ x; /Ix, di/ ~ 1 J . (2.31): (2. 31) follows from (3.32) and (3.46) (2.32 ): Use (3.34) and the fact that D is a strongly partially ordered set. Hence (2 .33): This is irr,mediate from (3.44). (2 .34): Follows from (3.42). (2.35): A direct result of (3.43). T is an N1 -space w.r.t. A, 7.1 and the first state- ment of the theorem has been proved. T be an N -space w.r.t. 1 Conversely, let A, U . Given u 0 lL define u *= [ ax; :k E U, / a/ f: 1] and ii 1r = [u~; u EV Also given U~ U J • Given U~ € 711'we see that uc u "• we have by postulate (2 . 33 ) of definition 2.31 that there exists a V € 11 such that V* C u ' and 4U U* is equivalent to 1J. . From this equiva lence hence l/. * satisfies it follows immediately that (2.31) - (2 .35) of definition 2.3i, and also satisfies: 1' ( 3.5i ) 3.52 ) ( /a/ f; i au*c U for given a. c A and U such that Pi V --y-- ~ and each "}} -jt 6. u. * 1 1* U E u. there exists + p2 V-¥ C U * for all /p 2/ ~/a/ • ... where For let us suppose that XE- aU and x = ay = apz x y = pz , where Ip/ ~ i and by (i.22 ) /ap/ /a/ f: 1 ; i.e. zeU • and = /a/ /p/ ~ 1 , and hence Therefore (3.5i) is satisfied by 71*. )( € U ,:. U*£ U * definition 2 .31 states that given "" 6 U* a Vi such that v1* + v1* c U -)(- • 'I J* -",c- (2 .34) of there exists Now by ( 2 . 33) of definition 2 . 31 we have t hat given _ a€ A -;4 Then there ex ist s a = 7' V E: u_ s u ch that pV c v for /p/ /a/ . Therefore 1 * ~ ~ ~ * PiV + P2V c. vl + vl c U for /p1/ f: /a/ and /p2/ 6 /a/. Hence (3.52 ) is satisfied by Uk. (It is clear that (3.52 ) holds also for 7} ~ 1 since we made use of only those properties of Ll ~ that are al so properties of Define //x,U-JI:// f: g.1.b. [!a/; . x t aU bound exists by (2 . 35) for eaGh. x e: T and J. 11 . ) Th is lower U * € 1l* , and 1/..x,u*// is ~non-negative real-va lued function defined on T 11 ,. set with . '1 J ~ u. u; ~ u is cle a rly a strong ly partially ordered 1 1 if u 2 c u; . of definition 2 .31. This follo ws by (2 . 32 ) ; 41 -:ll- In order to show that #x,U # , as defined above, is a pseudo-norm, we must show that #x,U *'// satisfies (3.32), (3.34) and (3 .41)-(3.44). (3.32): and all By definition #x,U~ ~ 0 for all x {:; T u* t= 11* . Let us as su me that #x,U// ~ 1 for u* ~ 11* . Then, given U x E U * and a EA , we have by (2 . 33 ) of definition 2 .31 that there exists a such that pV ~ c U * for all /p/ ~ /a/ . v* €- U ~ Pick /a/> 1 , which is possible by postulate (1. 2 5) of definition 1. 21. Then, since #x, v*§ .f l , we -have that there exist such that X f i. e., x E U * pV 7' /p/ < /a/ • , Hence U~ E l i-,t for every (2 .31) of definition 2 . 31 that U~ , x e pV*c It then follo ws fr om • x p EA = e , and ( 3 . 3 2 ) of defi- nition 3. 31 is satisfied by #x,U *#. (3.41): If x €: pU * , t h en ax t: apU -ft' , and since /ap/f /a/ /p/, we obtain from the definition of the function * ~ ~ #x,U '// that #ax,U // f: /a/ #x,U # (3.4 2 ): /a/ ) 1 , . We have from (3 .52 ), by choosing that given such that #x, V*I/ 'fl -L l UTE. u , ;1t there exists a §y, V~"// f 1 at. A v1t € U * i mp lie s that and .yE p v*, where /p / Z. /o/ and /p 2 / .( 1 2 and #x + y,U) ~ 1 . Hence (3. 42 ) is satisfied by #x,u *'// . (3.43 ): and x c- pl V * /a/, the It follo ws i mmed iately by/definition of ~ . * and (2.35) of definit-ion 2.3l that //x,U '// //x,u II satisfies this postulate. a V~ Ell* ae A , Given (3.44): such that • U.,x 6 U * , there exists u* , for all pV~C /p/ ~/a/. This is by postulate (2.33) of definition 2 .31. a0 be such that such that /a 0 / p0 V0* C v-r.. for all /po/ ~ /ao/ • * -L 1 , there then exists a /Ix, V// * , XE PoVo px £ pV* c: u* 11* If p 0 ~ A such that • Hence x6p-i/cv* and 0 0 , /p/ ~ /a/. I.e., given /po/< /a 0 / Vii'0 E Then t here exists a 1 . ) Let aEA U * c 11* and * ,, ~ such that /Ix, V // ~ 1 implies 0 * ~ 1 for all /p/ ~/a/. Hence (3.44) that //px,U'// of definition 3.41 is satisfied by /Ix, U*/,/ • there exists a (3.34): 7f VO E u.. * Ul C If Ul ~ u2 U ~ 2 and x €. aU -}Ir 1 , then x ~ au * , 2 implies //x,u // ~ //x,u2 // 1 for all Therefore we have shown that an N1-space T can be and hence xfT • 11 * A, 1.1*. pseudo-normed w.r.t. in such a manner that T is a P1 -space w.r.t. It remains to show that the set of all u(u*) is equivalent to = [ x; 1/x,ui; ~ 1) 11 . Given U(U;it-) it follows immediately from the definition of u(u*) that //x,ut Uc U(U~) • and the definition of Conversely, given U E:- 'lL 43 we have by ( 2 .33 ) of definition 2 . 31 that given v¥~ 1c~ such that there exists a pV * C a c: A u* for /p/ f /a/ • Pick a such that /a/) 1 • If /Ix, V *// f: 1 , then there exists a p c A such that xt: pV f.: /p/ ~ /a/ • and u(v*) c u* and //x,U~ is equivalent to shown that Therefore xeU * • Hence the neighborhood system generated by U , since we have already U* is equivalent to U . This completes the proof of the theorem. An important example of a P -space is the "topologi1 cal abelian group 11 considered by Michal in C CO] • We shall discuss this example in detail later in the paper. THEOREM 3.52. also ~-~•i• A, D1 , that given df= D there exists£ then£ necessarz and sufficient 1 condition that the t wo pseudo-norms be equivalent is /Ix, d // f //x,d '//' Proof: d'E- Dt such that and conversely. The sufficiency is evident. from (3.41) of definition 3.41. The necessity follows Suppose that the pseudo- norms do generate equivalent topologies, i.e., given such that /Ix, di/ f 1 d c D there exists a d' €:: D' all /Ix, d' // 1{:- 1 . Then, given 1 See def inition 3.12 , p . 29 . r > 0 , x ET , and for 44 d' E D' , there exists an x = ay , /a/ - //x,d '// 1 f: r a f. A and y €: T such tha t and //y ,d f// 1 !::. 1 . Hence //x,d/1 =//ay,d// f:/a.///y,d// f: /a./ =//x 1 d 1//' + r 0 L ro - wher e x c T . Therefore /Ix, d // f //x,d '//' r . , for all The converse is proved similarly. We are now able to shovv tha t we can start with a weaker set of postulates than those of definition 3.41 and yet retain the possibility of defining a pseudo-norm which will satisfy definition 3.41. COROLLARY 3.51. If T is g_ v1 -space }!.r_ • .t. A and D is£ ·stronglx_ partially ordered set, and if a realvalued function /Ix, di/ //x,d// is defined on TD §Jd..Ch that satisf ies postulates (3 . 31), (3 . 32 ), (3. 34 ), (3.42) , (3.44) and (3. 51 ) an a EA giveIJ. xt- T and and y t. T d ED such that there exists x = ay , 1/y, d // f:- 1 , theIJ. T may be pseudo-normed w.r.t. D in such~ manner T is~ i;,-space ~-~•1• and the new pseudo- that A,D ~ 1/x,d// 1 is equivalent to /Ix, d // . Proof: If we define U = [ U(d) ; U(d) = [ x; de D] , /Ix, di/ f: D and the above properties of are sufficient to prove that T is an N1 - s pace /Ix, di/ A, U . This corollar y then follows dire ctly from theorem 3.51. 45 CHAPTER 4 ~ -LINEAR FUNCTIONS ON P1 -SPACES TO ~-SPACES In theorem 3.52 we showed t h e equivalence of P 1 spaces and N1 - spaces with the pseudo-norm generating the neighborhood system. Hence we can study such spaces either as characterized in terms of a neighborhood system of the origin or in terms of a pseudo-norm. The pseudo-norm is, of course, a generalization of the norm, and for this reason we shall find it more sugg e s tive and also more convenient to work out our theory in terms of the pseudo-norm. It was shown in theorem 2 . 31 tha t eve r y N -space is 1 a topological abelian group, and h ence by theorem 3 .52 every P - space is a topological abe l i an group . 1 Corollary 3.31 will be of importance to us here a s it gives t he statement of to polo gical concepts in t he language of the pseudo-norm. The purpose of this chapter is to i ntroduce t he con ce pt of a linear f unction which poss es s es a s or t of gener alized homo geneity . We know t hat in t h e cas e of Bana ch s pace s that every linea r f un cti on (i.e., a f uncti on whi ch is both additive and continuous) is homogeneous of degree one. 1 Banach's proof of t his can also be ex tended to the ca se of linea r function s in linea r topolo gica l spa ce s~ Here we shall postulate a gener alization of t hi s homo geneity fo r 6 1 2 See l 3J , p.36 , theorem 2. See [ 9] . The pr oof i s f or a spa ce (F ) . 46 our linear fun ctions and shall choose this homogeneity to be such tha t we shall be able to obta in pro perties corresponding to the known properties of linea r functions in linear to pologic al spaces. l - Linear functions. 4.1. Throughout this se ction of t he paper , unle ss otherwis e stated, T will re present a P1 -space w.r .t. -a P1 -space w.r.t. on A to A', Dt , and such that A' DEFINITION 4.11. ~ A, D, T' (a) will denote a function / i} (a)/tf: /a/ An additive func tion 1 for all ac: A • f(x) T to on Tl will be said to be homogeneous with respect to a function p (a) on A to and x E:: T • We shall say briefly t ha t if At f(ax) = ~ (a)f(x) for all aE A f(x) is ~ TJ will additive . DEFINITION 4.12. be said to be A function f(x) on T to ~ - linear if f(x) is p -additive and continuous. DEFINITION 4.13. to Tl A i -additive function f(x ) will be sa id to be bounded if given exists a d = d(dt)E D for all xe T • on d'c D' T there such that //f(x),dl// 1~ //x,d// 1 A function f(x) is said to be additive if f(x) + f(y) for all x,y E T . f(x + y) = 47 N.~. Let f(x) be a i -linear function on T to T'. Then evidently 'p (m)f(x) = mf(x) ( 4.11 ) and all for all integers m x €: T , ~(a +f )f(x) = (~(a) + cf (f))f(x) a, f e A and for all x~ T , f (a p )f (x) = cf (a) ~ (f )f (x) (4.13 ) a, ~ E- A and for all x6 T , and = ~ (am)f (x) = ~ (ma)f (x) m~(a)f(x) . If we assume that plies that at= 0 or T' is such that yt = 0 1 , a'y' then if =0 im- f(x) # ~ , ~ represents the null function, ( 1.15 ) f (m) = m , ( 4.16 ) ~ ( a + f ) ::: ~ (a) +{Cf) , ( 4.17 ) ~(a f) = c£ (a) I (f) , ( 4.18 ) ~ (am) = cp (ma) = m ~ (a) , for all THEOREM 4.11. a a, (3 6- A and m any integer. A necessary and sufficient condition t ha t ~ -additive function f (x) be ~ -linear is that f (x) .Q.§_ bounded. Proof: The sufficiency follows immediately fro m Corollary 48 3.3l(c), since f(x) is additive. In order to prove the necessity let us assume that i f(x) is a -linear. / I (a)/t ~ /a/ f(ax) = ~ (a)f(x) , Then and given such that //f(x),dt//t 6 1 d' €:: D' for /Ix, di/~ l . statement follows from corollary 3.3l(c). //y,d// ~ 1 dG D , x e T , and exist such that x = ay , and y €:: T /a/ -//x,d//< r . x € T. and for all r x f:- T • r ) 0 there 1/y, d // 6 1 Then //f(x),dt//t =//f(ay),d'//' L /a/ = /Ix, di/ + ro, where any Hence By (3.43) of definition 3.41 we have that given and The latter implies that //f(ay),dt//tf= /f(a)/t ~/a/, applying (3.41) of definition 3.41. af A d = d(dt) there exists a r0 >r . Since this holds for > 0 , we have that /If (.x) ,d '// 1 -f: //x, di/ This completes the proof of the theorem. This theorem is then a generalization of Banach's theorem that any linear functional on a Banach space has a modulus. 1 DEFINITION 4.14. A set S c T will be said to be d G D the set [ /Ix, di/ ; x t::: bounded if for each S] of real numbers is bounded. N.B. If and Sl s2 are bounded subsets of T , then subsets of T • s u S2 and s1" s2 are also bounded 1 It follows from (3 .41) of definition 3.41 1 See Also it is clear that • and 49 that for any aS a{:. A and S is a bounded set. a bounded subset of T t hat Clearly any set with only a finite number of elements is bounded. A I -linear function COROLLARY 4.11. to Tt QD. T maps bounded sets into bounded sets. Proof: This follows immediately from theorem 4.11. A necessary and sufficient conditlon that THEOREM 4.12. g_ set F(x) Sc T be bounded is that given an Q.Q.fill set U containing the zero element there corresponds~ real number r = r (U) > 0 such that given x S there exists an a A 6 such that Proof: fied. I a/ -f: r . x 6 aU , Sufficiency: Then given E Assume that t he condition is satis- d€-D we have by corol1ary 3.3l(a) that there ex ists an open set containing the zero element and contained in U(d). Hence given xE S /a/~ r(U) and //y,d// Let U denote this open set. there exists an x e au c aU(d) . a <SA I.e., such that x = ay where =1 . Hence /Ix, di/ ~ /lay, di/ f: /a/ by (3.41) of definition 3.41. Therefore for any x GS, //x,d// ~ r(U) and the sufficiency is proved. Necessity: Let S be a bounded set and Ube an open set containing 8. Then by corollary 3.3l(a) there exists l Compare with the discussion on boundedness i n L~J • 50 a U(d) CU. By (3.43) of definition 3 .4l we have that given x€ S and y f: T such that r ) 0 there exists an ae- A and a 1/y, d // f 1 and / a/ - /Ix, d // f r . But now, since x t!: S and S is bounded, /Ix, di/ ~ r (d) . Hence /a/ I:- ro(d) + r . Therefore given x e S, x ~ aU(d) c aU, where /a/ f! r (d) + r for any r> 0 , x = ay , O 0 and this proves the necessity of the condition. Let represent the set of all Ff p -linear functions fl , f 2 E Fi , we define f3 = fl+ f2 to be the function • f 3 (x) = f 1 (x) + f 2 (x) • With this definition of addition we can prove the following on T to Tt • Then if theorem. THEOREM 4.13. Q.ll T to Proof: F£ of all ~ -linear functions is a topological abelian group. Tt It is evident that the sUJIL of t wo f -additive functions is A' . The set ~ -additive, since T' is a linear set over Also it f ollows immediately from corollary 3.31(c) and (3.42) of definition 3. ·41 that the sum of t wo continuous functions is a continuous function ., and hence f 1 + f 2 E F cf for every f 1 , f 2 E F ~ . The commutative and the associative laws of addition follow from the commutativity and associativity of addition in T' . 51 The null function ,& (x) = 8 in Ff then and Therefore f E. F ~ • f + ,.jL = f •for all - f , the function -f(x) , is in We wish now to show that sets contained in Let T • strongly partially ordered set with (Si , s2 E: BT • ) pairs w = (d•,s), Define BT be the set of BT is clearly a s1 ~ S2 and dteDt wl ~ w2 if i = l, 2 , • • • ) M(f.,w) = l.u .. b. [#f(x),dt/1 1 ; xt w = (d' ,s) E W M(f,w) is defined for each and f e F ?E • fc F'! and BT w is evidently ( w. = (d~,s1 ) J. where Sl ::> s2 • Since S6BT. are strongly partially ordered sets, a strongly partially ordered set1 with Sl ~ 82 • if to be the set of all ordered W Dt r f + -f = ~. F~, and is a topological group F! and we shall do this as follows. d1 ~ d'2 and Define f 6 F !f, If is an additive abelian group. F! all bounded x ~ T is contained for all s] , By corollary 4.11 and M(f,w) wcW. then has the following properties: (4.19 ) M(f,w) =: 0 M(f,w) !fl for all (4.110 ) given M(f 1 ,v) f 1 and for every W'=: W Wf: WEW implies that W there exist IvI(f 2 ,v) 6: 1 . f E F~ , and , f = ,.&. VE; W such that implies that M(f 1 + f ,w) ~ l ; 2 ( 4.lll) l See VI. M(atf,w) I.. /at/r M(f ,w) for each wE W 52 and each a'f'= F~; a Vf::: W for all such that /pt/t ( 4 .113 ) for each o.'e A 1 and w (:; w .t here exists given (4.112 ) wl ~ M(f.,v) f l /at/t ~ w2 IvI(plf ,w) p ' f !: F~; M(r,w1 ) ~ M(f,w2 ) implies that f e: Fi . These properties of of and implies that M(f,w) follow fro m the definition M(f,w) and the properties of //f(x),d'//' • M(f,w) as defined above clearly satisfies (3.31) F~ (3.34) of definition 3.31 and hence w.r.t. W • space. It then follows from corollary ~.3l(c) and (4.110) By corollary 3.3l(a) F~ is a P- group is a topological that the operation of addition is continuous. Since lf = f and -lf = - f E: F~ , it follows fro m ( 4 .111) and (1.24) that M(-f ,w) = M(f ,w) Hence by corollary 3.31.(c) tinuous and for all f €: F ~ and w6 W • the inverse operation is con- Ff is a topological abelian group. Later on in the paper we shall return to the study of such a space FI. and w~ th stronger conditions on we shall be.able to show that FI is a At P2 -space w.r .t. A', W. In preparation for this we shall now show that if is a commutative ring, then At w.r.t. A' • FI is a v -space 1 I.. 1 53 THEOREM 4J.4. If Q.ll ~ P1 - space T ~.r.1. A', Dt where F~ f -linear functions is the set of all A, D to~ P1 - space Tt ~-~•1• is a commuta tive 1 ring, then the topologi- Al Ff is ~ v1 -space with respect to cal abelian group We have shown in theorem 4 .13 that Proof: logical abelian group. For atf atf(x) . l t for each 1 a 1 a . €: A FI is a topofcF~ we define Since A' is commuta- a{f (x) + a~f 2 (x) 1 I and f € is f to be the function t-ive, it follows that and a'€:A 1 A' . f , F i -additive It follows from . 2 2 1 (3.47) of theorem 3.41 and corollary 3. 3l(c) that I I a 1 f 1 (x) + a r (x) 2 2 pick al to be is continuous in Xe or In using (3.47) according as f or /a1 / < /a 2 / and fl , f 2 c. F cf • One may then go on and easily verify that f • ' • f /a1/ ~ /a / 2 Hence F f is a v1 - space. 4.2. Sequences of ~ -linear functions. The first theorem of this section depends only upon T being a P-abelian group which satisfies, in addition, the stronger for m of postulate (3 . 33 ), namely, (3. -42 ) of definition 3.41, and the following postulate: ( 4.21 ) to each d E- D there corresponds an p= f a for all a, e 1 I.e.~ a /3 € A • et D /Ix, el/ 6: l such that implies that /1-x, di/ f: 1 . It can be shown by use of corollary 3.31 that this is an equivalent form for the definition of a topological abelian group. of This equivalence is made clear by § 16 and § 17 l 1~ 1 , pp.. 52 -57. THEOREM 4.21. ll 1 Let T 1 are sequences of elements of xn} and Yn} it follows that: (a) be a topological abelian grou~. lim x if T, exists, then the limit is n unique; (b) lim (x n lim xn =x and lim Yn = y im:glies .:the.:t. + y) = x + y. n If Tis a P1 -space ~-~·1• A, D, (c) lim x n =x implies that for any a6 A lim ax =ax . n Proof: As we mentioned above, every topological abelian group can be characterized in terms of the pseudo-norm. We shall give. the proof of this theorem using the pseudonorm. (a): Let us assume that lim Xn = x and Then by (3.4.2) and (4.21) we have that given exists an eG: D such that /Ix - y, di/ ~ l lim x n = z. dE D there for all //x, el/ L.. 1 a.nd //y, el/ f:: l .. By corollary 3.3l(d) there exists a positive integer m = m(e) such that for n) m(e) 55 //xn - z,e// ~ 1 . /Ix - xn., el/!: 1 and /Ix - z, di/'= 1 for all of definition 3.3l d6 D • Hence Therefore., by (3.3 2 ) x - z = e., or x = z., and the unique- ness has been proved. (b): Assume that lim x n = x and limy n = y _. Then, as in the proof of (a), we have that given there exists an eE D such t ha t implies that /Ix - y, di/ ~ 1 . and corollary 3.3l(d) //y,e// ~ 1 //x,e// f: land Then by our hypothesis there exists a positive integer ID= m(d) such that//¾_ - x, e//f: 1 for all n> m. all n) ID. dE D Hence and //Yn - y,e// ~ 1 //Xn + Yn - x - y,d// ~ l for By corollary 3.3l(d) this completes the proof of (b). (c): By hypothesis and corollary 3.cl(d) we have that eeD given implies that there exists an /Ix - xn., el/ .:f 1 . 3.41 we have that given an e €:: D m(e) such that at: A such that By (3 .4.4) of definition and d €:: D //xn - x, e // f:.- l //axn - ox, di/ f: 1 . Hence for n) m there exists implies n )ID(e) = m(e (a.,d)) we have that 1/a.x - axn, d // f:- 1 , a.nd by corollary 3.31(d) liID °'Xn = ax • COROLLARY 4.21. then Proof: Thus (c) If lim f lim f n(x) = f(x) has been proved. n = f for all , fn€.F'£ §.nd f€F§, X ET • As we mentioned at the beginning of this section., 56 every topological abelian group is a special type of Pgroup. Hence by theorem 4.13 and corollary 3.31(d) we have that given m = m(w) w€ W such that Since given M(f - f,w) ~ l n for all n) m. 1 there exists an SEBT such that x (; S , we see by the definition of M(f, w) that given d'€ D' xf:T there exists a positive integer m = m(w) , w = (d',S) , there exists an integer such that n) m implies that #'fn(x) - f(x),dtfft6- 1 . lim fn(x) = f(x) Hence by corollary 3.3l(d) for each X f T • THEOREM 4.22. fn (x) €: F~ , If lim fn (x) and if given = f (x) d' c.. D' for each xeT , there exists g_ d c D §!ill. g_ positive integer s = s(d',d) g'f 5 (x) - f(x),d.t§tf: 1 for cl-11 #'x,d#' ~ l , !hen is f -linear, i-~•, f c Fi Proof: Since such that . lim fn(x) 1= f(x) , it follows from theorem 4.,13 and theorem 4.21 that f (x) is f -additive. By 3.41 of definition 3.41 we have that given there exists an e'€Dt #'fn(x) - f(x),e'#' -f=- l d'6 D' such that #'f(x).,dtfft ~ l and #fn(x),e'#'' f:- 1 . Then, since is in that given F ~ , we have by theorem 4.,41 W and M(f,w) are defined in theorem 4.13. for Pick n = s = s(e•,e) as in our hypothesis. l f(x) f s (x) e' €: D' 57 there exists an all /Ix, e1// for 1 such that /If 5 (x) ,e 1// ~ 1 for 5 ., Then //f(x),d'//' ~ 1 1 By corollary 3.,3l(c) ·r(x) is Pick . /Ix, di/ f. l continuous. ~ D = 1 .. Now by hypothesis //f (x) - f(x),e'#'~ 1 /Ix, e /I ~ l for all e . d ~ e, e We have already shown that additive and hence f(x) is i -linear. f(x) is P- 58 , CHAPTER 5 f -DIFFERENTIALS OF FUNCTIONS WITH ARGUMENTS AND VALUES IN P1 :-SPACES. T, Tt and T'' will A,D ; A', D 1 and A'' ,D'' Throughout this chapter represent P1 -spaces wer.t. respectively, unless otherwise sta.ted. a function on A to A' such that all a EA • 5.1., ~ - approximation functions. DEFINITION 5.11., open set be a s8 c Let F(x) Then corresponds a ( 5 .11 ) e eD /~(a)/' ~ /a/ F (x) ~ -ru;rnroximation function 2n there exists an will denote for be a function defineq on an e ~ s8 • T , .P (a) s8 c such that given will be said to T to d t ~ D' T' if there d = d(dt)£ D for which: /Ix, e /I f: 1 and //x,d// f: 1 implies that //F(x) 1 d t//t f: 1 ; and ( 5 .12 ) /lax, e II f:. 1 that // ~ (a) F (x) 1 d ' // t N.B. and /Ix, d // "- 1 implies = l ., We have by corollary 3.3l(a) that there exists a U(d1 ) c s8 , and it is implied in definition 5.11 that e ~ d 1 , as we can only speak of (5 .. 11) and (5.12) being satisfied for such x €: T for which F(x) is defined. 59 1f. F(x) is both §:, LEMMA 5.11, an9- §:. ~ -approximation function Q!l =8 F (x) ~ -filigitive function fQ.!:. all X 6 T T T' , then iQ. • By hypothesis we have that there exists an d'E D' such that given et: D d = d(ctr)c D there exists a such that: /IF (ay) , d '// ' = // I (a) F (y) , d ti/ ~ l ( 5 • ]_3 ) for all /lay, e/1 b l and /ly, d/1 f:-- l • In addition, we have by (3.43) of definition 3.41 that given ;x ~ T such that and there exist d€ D x = ay y E- T and a(:; A Hence, if /Ix, e /I ~ l , and /ly, d/1 ~ l • we have from (5.13) that /IF(x),d'#' ~ l ( 5 .14 ) for all Hence by {3.32) of definition 3.31 #x, e# f: 1 c F(x) dtE Dt • =8 for all Then by use of (3.43) of definition 3.41 we have that given x €. T there exist such that #y, e/1 ~ 1 and F(x) is ~ -additive., F(x) = x = ay • a 6 A and ye T Therefore, since m(a)F(y) = 8 for each xe T, and the lemma is proved. DEFINITION 5.12. Given a valued ring Ao= [a;ac.A, ap = pa A define for each We see that given any integer m that p eAJ m= ml e A0 • ., 1 l ma=~+ ••• (m su..rnmanas) e•• +a. A0 can easily be shown to be the largest subring of A which is commutative. oU We are not in general interested iri the open set on which a ~ - approximation function F(x) is defined and so we shall not specify the open set but say onl y that F(x) is a f - approximation function defined on T to T' , though it is to be remembered that be defined on an open set s9 c T , F(x) may only e eS 8 • LEMMA 5.12. l - approximation on is~ ' ' ' a2E(A1)0 al, T to TY for all to be either ai or • at ' depending on a2 Proof: Pick whether /a{/ t ~ /a;/' or /a~/' > / a1' /' . by (3.47) of theorem 3.41 tha t given d' E Dt such that //F(x),d{//t ~ 1 Then we have ' d c D1 there exist 1 I for /IF l (x) ,d '// f l and §F2 (x), d t// 1 f= 1 . Since F1 (x) and F2 (x) satisfy exist (5 .11), we see that there A ei e D such that given d' ~ D ' there correspond d. c D such that J. //Fi(x),dt//t :f- 1 //x,d/f f:: 1 , e ~ e1 , e 2 i = 1,2 . 1/x,e/ f: 1 for all Hence picking we have that given ' and d ~ d1 , d 2 and d t:. D• 1 /IF (x) ,dl_// •f: 1 for all /Ix, el/ f: 1 and /Ix, di/ ~ 1 . Since a{, a~ + a.~ cp(a)F2 (x) • can be shown that are in (At) 0 , Q? (a)F(x) = a{ ili (a)F1 (x) Then in a manner sin~ilar to the above it til. /I f (a)F(x),d{//'_ b 1 /Ix, d// f: 1 . for all //ax, e// ~ 1 (It is to be noted that e and and d may have to be rechosen in order to hold for both this equation and the one above. partially ordered set.) Therefore and (5.22). is defined on If F. (x) 1 i Se C s1 n s2 , , LEMMA 5.13. If 2!l T Tt to to Hence F(x) F(x) is~ is ~ is a such i -approximation I , then G(x) = g(F(x)) to T1 ' Since 1 p -linear function Qil g (x t) T 86S ~ -,approximation function and Ttl T , T' • T' function Q!1 Proof: to S. C since by corollary 3.3l(a) an open set does exist. T satisfies (5.11) is def·ined on an open set F(x) 8 e s8 • function on F(x) l. = 1, 2 , then and D is a strongly This is possible since g(xt) , is a where p -approximation .p (a) = f ( ~ (a)) • is a ~, -linear function, we have by theorem 4.11 that given d'' E D'' there exists a d 1 e D' such that ( 5.15 ) //G(x) ,d' '//' t ~ #F(x) ,a t,f t Because pf the fact that F(x) is a • ~ -approximation function, (5.11) is evidently satisfd.ed by G(x) .. By the if -linearity of ( 5.16 ) Now G(x) g (x') 'f (a)G(x) = g ( rp (a)F(x)) . satisfies (5.15) and hence ( 5.17 ) # P (a)G(x),d"#" b ,fp(a)F(x),d'#'. G(x) t hen satisfies (5.12) by virtue of the fact that I P (a)/" = / ! (a)/t = /a/ . F(x) does. Clearly G(x) is a f - approximation function on T to T Hence t ' , G(x) is defined on the same open set in T on which where F(x) is defined. 5,.2. ~ -differ entiable f unctions .. DEFINITION 5.21. set Sy c T to Let f(x) , y e Sy . Tl f -linear function be a ·function on an open f (y; ox) If there exists a of ox on T .• to such TJ that ( 5.21) is a F(ox) = f(y + ox) - f (y) - f(y;ox) f -approximation function of ox on an open set e e. s T' , then f (x) ¢ - di fferenti aQk at x =y and f(y;ox) a ~ -different ial of f(x) at x =y Se c T , N.. B. 8 , to will be said to be will be called with increment ox* There exists by corollary 3.3l(a), coroll ary 3.3l(b), and lemma 3. 31 an open set y + s F(ox) 6 c Sy • THEOREM 5., 21. y f Sy , to is f (x) s8 c T. to on §.!1 open set ~ -differentiable f(y;ox) ~ ~ - differential of uliligue e e f-' s8 , such that i s then a function on If g_ function T' s8 c T , at f(x) , then x =y S y with_ f(y;ox) is T-' • c T , 63 froof: Let us assume that are both i f 1 (y;ox) -differentials of f(x) r 2 (y;&x) x = y. Define and at G(ox) = r 1 (y;ox) - f2(y;ox) • Since G(ox) = ( f(y + ox) - f(y) - f (y;ox) ) 2 ( f(y + ox) - f(y) - f1(y;ox) ) , we then have by hypothesis that G(ox) of two f -approximation functions. G(ox) is itself a to Tt and by theorem 4.13 on T to for all tial Hence by lemma 5.12 ~ -approximation function. G(ox) ~ -linear functions is also the difference of two Tt . is the difference G(ox) is a ~ -linear It then follows by lemma 5.11 that ox E. T , and thus the uniqueness of the on T function G(ox) = e g5 -diff eren- has been proved. THEOREM 5.22. If f(x) at _ x = y , theu f (x) is~ i-differentiable function · is continuous at x = y , .. Proof: By (3.42) of definition 3.41 we have that given d t f: D' there exists an e 'E D' such that //f(y + ox) - f (y) ,d t//t ~ l for all //f(y + ox) - f(y) - f(y;ox),et§t ~ 1 //f(y;ox) ,et§t is a =1 • Since and f(y + ox) - f(y) - f(y;ox) p_ -approximation function and f(y;ox) is a linear function, it follows that there does exist a such that #x, d# ~ 1 implies that 4) d~ D 64 II f (y + ox) - f (y) , d 'II ' .(: l Therefore by corollary 3.3l(c) f(x) . is continuous at x = y, and the theorem is proved. y ~ Sy , QQ an ~ -differentiable is oDen set S c: T y at x = y , if §Pd ~ ' -approximation function on an open set is§:. to f (x) If LEMM.A 5.21.. Ttt 1 e~s~, then H(x') S~ c T' G(ox) =H(f(y + Qx) - f(y)) is s8 c: T to g_ (J! -approximation function on an open set 1 'f (a) = ~ ( t (a)) • T• t , e e s8 , where Proof: Since T' T 1 ' , we have by (5.11) of definition 5.11 that to there exists an corresponds a H(xt) et€ Dt d'£ Dt is a ~ ' -approximation function on such that given d 11 E D1 ' there such that II G(bx) , d , '// t t f: 1 for all // f (y + bx) - f (y) , e '// ' L 1 and // f (y + ox) - f (y) , d ti/ ' ~ 1 . By theorem 5e21 and corollary 3.3l(c) there exists an e E. D 1 and d G: D 1 such that //f(y + ox) - f(y),dt//t ~ 1 and for //'6x,e/ ~ 1 . 65 Hence G(ox) satisfies condition (5.11) of definition 5.11. By (5.12) we have that there exist e•~ Dt given dtte D'' there correspond d'~ D' (5.22 ) for such that such tha~ // Y (a)G(ox) ,d' 1// 11 f: I //f(a)f(y + ox) - f(y)),et//rf: 1 // f (y + ox) - f (y) , d t§ ' ~ 1 and a By theorem 5.12 and corollary 3.31(c) there exist ct1 E: D such that (5.23 ) //f(y + ox) - f(y),dt// ,t ~ 1 Define F(ox) ~ f(y + ox) - f(y) - f(y;ox) . F(ox) is a , ~ -approximation function. e ' f Dt 1 ( 5.24 ) for //ox,df/ ~ 1 . By hypothesis It then follows from (3.42) of definition 3.41 that given exist for e'E D' there such that //~(a)(f(y + ox) - f(y)),e'//' f: 1 //~(a)F(ox),el_//' ~ 1 q.11d Due to the fact that F(ox) //~(a)f(y;ox),e{//' f= 1 . is a ~-approximation function, we have by postulate (5.12) of definition 5.11 that there exists an there corresponds a \ 5.25 ) for e ED such that given 2 d f D such that 2 I e t D1 1 // f(a)F(ox).,el' ~ 1 //a(ox) ,eif f:: 1 and //ox,d.j/ ~ l Since, by hypothesis, f(y;ox) is that there exists an e c D 3 such that . ~ -linear, we have 66 11 ~ ca)f(y;ox),e:f' = 11r(y;aoxt,eE' f:, 1 ( 5 .. 26 ) for 1/aox,e.j/ !::: 1 • Hence by (5.22) - (5.26) we have for that given e 4 ~ e there corresponds d d t t E. D" 4 , e 2 3 ED 2 d , d E D 1 2 such that // p (a)G(ox) ,d t ti/'' ~ 1 \ 5.27 ) for 1/aox,e / ~ 1 //ox,dif ~ 1 . and d =d Therefore for G(ox) , and G(ox) satisfies definition 5.11 s8 CT ~ -approximation function on an open set such that y + S 8 C before, such a s et does exist. we have that 4 to s is a Tt ' is S8 • As we have shown y • Clearly / P(a)/" ~ / p (a)/t I:= /a/ . This completes the proof of the theorem. s1 c sets T and to T' , and if at X tiable f (x) If THEOREM 5.23. s2 c f(x) ~ function and g (x) respectively, ye s T and g(x) t at f , s 1 2 , ~ -differentiable are t h(x) = a f(x) + a g(x) 2 1 ~ = y for any a 1t , a 2' c A''? = y, then on open is - ~ -differen- with f h(y;ox) = a f(y;ox) + a g(y;ox) . 1 2 Proof: By lemma 5.12 h(y + ox) - h(y) -a ' f(y;ox) - a f g(y;ox ) 1 2 is a ~ -approximation function. As in theorem 4.14, t t . h (y; ox) = a1 f (y; &x) + a 2 g (y; ox) and the theorem is proved. if: is a :±: -linear function 67 THEOREM 5.24. to Tt Q!1 an x = y and if is Zf5 -differentiable at s} (y) c T v open set Sy c T , y €; Sy , f (x) Q.!!. .fill open set If g(xt) T' ' , f (y) € sf: (y) , to is f -differentiable at x' = f(y) , then h(x) = g(f(x)) I QJl aye T to is ]! -differentiable at x = y , Tt' 1!" (a) = !' (<J2 (a)) where Proof: h(y;bx) = g(f(y);f(y;ox)) . and Define G(ox) = h(y + ox) - h(y) - g (f(y);f(y;ox)) Then G(ox) \ 5.28 ) = g(f(y + ox)) - g(f(y)) - g(f(y);f(y + ox) - f(y)) + g(f(y);f(y + ox) - f(y) - f(y;ox)) because, by hypothesis, g(y•;ox•) y' = f(y) • Since g(xt) is additive. Here f -differentiable at x' = yt , 1 is we have that H( ox ' ) = g (y t + ox t ) is a g (y t ) - f ' -approximation function on by lemma 5.2l H(f(y + ox) - f (y)) is a o/ -approximation g(f(y);f(y + ox) - f(y) - f (y;ox)) 1~',\tH. . ,. _ -approximation~in ox on Ttt . Tt t 5.13 ~ to to ox TTr T Tt function in that on g (y t ; ox ') - T to • Then It follows from lemma Ttt . is a Since G(ox) = H(f(y + ox) - f(y)) + g(f(y);f(y + ox) - f(y) - f(y;ox)) , it then follows that mation functions. G(ox) is the sum of two Hence by lemma 5.12 G(ox) 1£ -approxiis a 68 ~ -approximation function on T to T'' . It follows almost immediately from theorem 4.11 that h(y;ox) = g(f(y);f(y;ox)) is a -P -linear function on T to T'' , and .the theorem is proved. ~ - differentials in Michal's "topological abelian g__roups . 11 1 5.3. A. D. Michal has considered the notion of a differential for functions on an open set of a tttopological abelian groupn to a "topological abelian group." section of the paper to show that the We wish in this 11 topological abelian groupn which Michal considered is a P -space and that l every function which is M -differentiable1 is differentiable 1 in the sense of this chapter, the two differentials being equal. By a 1~opological abelian gro@"(i.~•&•) is meant a 2 topological abelian group T which satisfies the following additional postulate: given 5.3t x~ T and any open set 8E:U , there exists a positive integer lt 6 nU . We shall always refer to such a topological abelian group as a t.a.g. 1 2 See [j DJ See XIII. • n Uc T , such that THEOREM 5.31. Every i•~•K• is fill N -space with respect T 1 11 , where A is the integral domain of all j.ntegers with the absolute value .§:..§.. the valuation and 11 = f-u f iQ. A, is the set of all ooen sets U of T containing 8. versely every N1 -space with respect to Con- A, 11, where A is the absolute-valued integral domain of all integers, Proof: It is evident that a t.a.g. is a v1 -space with T respect to the integral domain of all positive integers with /m/ = jml Also it follows from the definition • of a t.a.g. , theorem 2.23 and theorem 2.24 that: the intersection of al:l given a W E- U such that given such that and V €- is 8 ; 11 there exists Wc U " V ; U € 11 there exists a V E ti. u E-11._ there exists a V ell -V c U j given such that U € 1l U E 1L V +Ve u ; given a positive integer Xc;T and UE'U there exists m such that . x t mu ; where 11 is the set of all open sets of T containing e. Hence (2.31), (2.32 ), (2.34) and (2.35) of definition 2. 31 are satisfied by U . Novi we have by (5.34) that given a V E-1.l such that Ut lL there exists -V C U or what is equivalent V C -U .. w €:: By (5.33) there exists a wcu, -WC U .• Hence -U n U • (5.35) we have that given such that V +Ve u Hence given UGll Vn €- 11 such that and 2n ~ / m / u such that W c V n U c • U El/._ Then by the above and there exist VE: -u._ , V -VCU and -V - V C U . and m any integer there exists a ±Vn ± ... (a" summands) •.. ±Vn c U Hence for \ m1 \ ~ j m\ 6: 2n ; m1 Vn c U . Therefore (2.33) of definition 2.31 is satisfied by 1l, and the first part of the theorem has been proved. The converse is an immediate result of theorem 2.31 and theorem 2.24. It then follows from theorem 3.51 that: COROLLARY 5.31. 11, where Every P -space T with respect to A, 1 A is the absolute-valued integral domain of all iptegers, is~ t.~•K• Conversely every t.~.~- ~ £.fill be pseudo-normed with respect to~ strongly partially set ~ D in such~ manner that the pseudo-norm generates neighborhood system equivalent to~ complete neighborhood system of the zero element, and respect tg N.~. T is~ P1 -space with A, D. It follows from (3.43) of definition 3e41 that the pseudo-norm whi ch may be defined for a t.a.g. T (as in theorem 3. 51) is integral-valued and that /Ix, di/ = O 71 for any x = e . implies that df D By the above corollary all the theory that we have developed for P1 -spaces is applicable to Michalts "topological abelian groups." • Example~ We shall now show by an exa~ple that every P1 -space We know that if w3r.t .. A, D is not a t.a.eg~ is a T P1 -space w.r .. t. .. A, D 11 then T is a topological abelian group, a.nd though it is true that given x E T and U(d) ~ U there exist a. 6 A such that x ~ aU (d) , this does not imply that t:bere exists an integer m€: A with the above property. Let r €: Rp R p be a p-adic number field. the valuation of /r/ = pw 1 ·Then given r , /r/ , is defined as follows , where O <.. p c._ 1 and is w the order of r . This valuation can be.shown to be a non-trivial., nonarchimedean valuation of R , tl1e field of rational numbers. r, r e: R ' i = 0,1,2, ... i p /0/ = 0 ,, /r/) 0 r t 0 ( 5.37 ) I.e .. , fer each 1 . ,. , ,• ( 5.,38 ) /rlr2/ = /r1/ /r2/ ( 5.39 ) /rl + r2/ f:: max( /r1/ and /r 2/ ) ( 5.310) /r 0 /to, 1 ( 5.311) /n/ ~ l for some r 0 ER for all integers ., ,. n • See li} , pp" 289-292 P for the definition of R and all tbe properties of p-adic number fields referred ¥0 here., 72 Rp is clearly a valued ring., and we shall show that R, R+, where p positive numbers. :r_eal with respect to RP, and set with ordering as usual. p is the set of Evidently R+ R is a v -space R 1 p is a strongly''partially ordered Define and o>0 • It follows irnmedia.tely from (5.37)-(5.310) that /Ir, '6// /Ir, '6/1 = o /r/ for each r E RP satisfies (3.31), (3.32), (3.34), (3.42) and (3.44) . From (5.38) and (5.310) it follows that given E> 0 such that there exist and /r/ >f ' r e Rp , r # 0 , x = r (r-lx) ' . we can pick for any given such that '6 €:. R+ Given • x c Rp Then from the above and x E-R an p re R //r-1x, o// ~ 1 , i.e., /r/ ~ /Ix, o/1 . Hence //r,d// satisfies (3.51) of corollary 3.51. Therefore b;r corollary 3 .51 Rp, R+ ,. is a P1 -space w.r.t. RP It follows also, since the valuation of medean(i.e., /r/ oc R+ and n E: RP with the ·property that For if we pick Now R p is non-archi- satisfies (5.311)), that there exist x E RP /Ix, '6/1 p x such that there exists no integer and '6 x = ny and //y, '6// ~ 1 . to have the ~roperty that > 1 , then 1 </Ix, '6// = 1/ny, '6// = /n/ //y, "6// • O < #y, bf f: 1 requires that /n/) l .. But by (5.311) no such integer exists. Therefore we see that we have proved the following 73 theorem. THEOREM 5.32. Let be any Q-adic number field and RP be the se.t. of QQ.§..il_i ve real rn.1mb.~u. le.t_ R+ Th.fill Rp in the sense of Michal. THEOREM 5.23e Every additive an<l continuous function Qll a i•~•K• T to~ i•a•K• T1 is a Qll T ~ (n) =n for anx_ integer n . to T1 where ~ -linear function Proof: The proof of this is clear. In what follows T, Tt and T'' will be tsa.g. 1 s , and we shall deal with them as P1 -spaces pseudo-normed with respect to D, D', and Dtt • The definition of an 1 Mi-differential which follows is that of Michal translated in terms of the pseudo-norm. Let DEFINITION 5.31. Sy c T to T' , y €: Sy • f(x) be a function on an open set If there exists a function f(y;ox) l 5.312 ) is additive and continuous in ox on to T there exists a function with arguments in T and values in (a) £ (y,e.,x) = e 1 See • T' which €(y,x 1 ,x 2 ) such that for all x ET ; 74 (b) there exists a d E D such that l (y,x1;nx2 ) = m £ (y,x1 ,x2 ) integers //x1 , di/ f: 1 m , (c) and all there exists an d' e- D• given there exist //£ (y ,x1,x 2 ) ,d '//' !:= l for all positive x2 € T ; e €: D such that d GD such that for //x 2 , e II f: 1 and //x1 , d // f- 1 ; there exists an open set s8 c T , e G S8 , such that f(y + ox) - f(y) - f(y;ox) for all f(x) then = £(y,ox,&x) ox £ s8 ; is s aid to be M 1 -differentiable at is said to be an M1 -differential of f (y; ox) x = y with increment If Sy c T to and if f (x) x = y , then f <x) =y • f(x) at ox • THEOREM 5.34. Tt X f(x) is is~ function Q!1@ open set is M1 -differe:ntiable at ~ -differentiable definition 5 . 21, where ~ (m) in the_ sense of = m for each integer m. In this case we have also that the two differentials~ Proof: In order to prove this theorem we need only show that F(ox) = s (y' ox, ox) with f (m) = m • 5.33 that an M1 -differentia.l in ox . is a {f -approximation function This follows since we know by theorem f (y; ox) is 4?_ -linear 75 We have by (5.313 (c)) that there exists an such that given d' {;; D' #F('6x) ,d '#' f:- 1 for therefore satisfies (5.11) of definition 5.11. F(ox) there exist e I!: D //bx, e II ~ 1 de:: D such that and 1/&x, d# ~ 1 .. By (5 ,.313 (b)) #mF(ox) ,d tg' 1 = # E_: (y ,ox, /ml ox) ,d '# t for all //bx, d# f:- 1 and m any integer. • Hence by (5.313(c)) //mF ( ox) ,d ti/ t f: 1 and //·'6x,d// I: 1 , satisfied by and F(ox). // \ml ox, el/= #mx, el/'=· 1 (5.21) of definition 5.11 is Therefore f -approximation funct'i on. uniqueness of the for [ (y, ox, ox) Since we have already shown the I-differential, if it exists, must equal the the proof of this theoreme is a the M -differentia.l, 1 § -differential. This completes 76 CHAPTER 6 PSEUDO-NORMED LINEAR SETS OVER VALUED DIVISION RINGS. 6.,1. P2 -SPACES. P2 -!rn_aces. DEFINITION 6.11. A v1 -space to be a ~ - spac~ ~-~•1• Let DEFINITION 6.12. T w.r.t. A if E A will be said A is a valued division ring. 1 be a strongly partially ordered set with positive real numbers associated2 and space w.r.t. Then A • with respect to T a V2 - T will be said to be pseudo- normed if there exists a real-valued function E /Ix, e /I on TE which satisfies the following postulates: ( 6.11) /Ix, el/ J 0 for all x~T and e E: E ,. //x,e/1 = 0 for all e E: E implies that X = (6.12} 1/a:x., el/ = /a/ /Ix, el/ for all a E:. A, x E T and e ~ E ; //x,re// = r /Ix, e /I for all X E:, T and re E: E (i.e., r > o and e e; D or e E. E ) ,. {6 .13J given e E- E there exist (,:;I • V ' d EE such that /Ix + y, e // f: /Ix, d // + //y, d // ; (6.14) for all e ~ d See II. 2 See VI . /Ix, el/ ~ /Ix, di/ X E:. T • We shall say briefly t hat 1 implies that Also T is a and ~-space ]!.!:,.t. • A, D. 77 LEMMA,6.11 .. A necessary and sufficient condition that la ~I = lal I (3 I A is that Proof: for all la/ la-l.l = 1 a, (3 ill g_ valued division ring for all llim_-~ elements The necessity is immediate since implies that a-/- 0 • Assume that /al la- 1 1 = l /a ~ I f : /al / pl By hypothesis = lal /~/ la(3/ = /11 = l . lal la.-l/ = laa-l/ Sufficiency .: a f A • for all for all a., (3 E: A • Hence and Therefore and THEOREM 6.11.. A Every Pl-soace T ~•£.•i• is fa. valued division ring such that for a l l a, (3 E. A , i~ a A.,D, where /a f l ~-s12ace Y!•:£.•i• to = /al / ~ / A , E = R + D • Conversely, every E -space ~.r_.t. A, E is g_ f -§J1ace 2 1 yy_.r_.t. A, E with /a. ~/ /a/ I <3 / for all a, (3 0 A , = and the ~ pseudo-!12.r!!l Proo_f_: For the first part of the theorem we have by hypothesis that 6.11 /Ix, e /I it is egui valent to /Ix, e /I . /a (3 / = la/ I [6 1 /a/ /a- 11 = l for all and hence by lemma. a i O. Then by (3.41) of definition 3.41 //ax, d // L /a/ /Ix, di/ , and /Ix, di/ f:: /a-1/ //ax, di/ or Therefore and /a/ /Ix, d // -!:: /lax!! d // • /lax, di/ = /a/ /Ix, di/ for all a E:. A , xE T d E: D , since the equality evidently holds for a =0 • Define E = R+D = l rd; d D , r > 0 ] E and //x,rd// = r /Ix, d // • Then E is a strongly partially ordered set with e 1 = ridi (i = 1,2) , if 1/x,e{I = 1/x,e/ for all x (:; T . for all d GD if r 1 (r 2 d) = (r 1 r )d 1/x,ef! ~/lx,eif, Hence and ld = d, and r , r 2 '> 0 • 1 It should then be clear by the definition of /Ix, e /I 2 for e E. E that /Ix, el/ of definition 6 .. 12. and satisfies (6.11), 6.12) and (6.14) Therefore in order to verify the first part of the theorem we need only show that /Ix, e /I satisfies (6.,13). By (3,.42) of definition 3.41 we have that given there exists a implies that a1 ~ D such that /Ix + y, di/ f:. 1 . d E:. D //x,d-j/ f l and //y ,d/ b. 1 Assume that //x,d-j/ .~ //y ,d-j/ 79 and /Ix,~// IO. have that given Then by (3.43) of definition a.41 >0 r there exist a EA we such that 0 ~ /a/ r f: r , 0 and //a-l y ,dfl L- l • Hence //a- 1 (:x + y) ,d// f 1 or /Ix + y, d f: /a/ ~ 1/x,d-JI + r for any r >0 • x, y i:: T and any Therefore /Ix + y, di/ f Max ( //x,d/ , //y ,d-f/ ) I:. //x,d/ + //y ,d{/ Given for e = rd we have that E . /Ix + y, el/ f //x,rdf/ + //y,rd{/ rd ~ E, and the proof of the first part of the·theorem 1 has been completed. T be a P -space w.r.t. A, E. It 2 is evident that //x,e// satisfies (3.34), (3.41), and (3.42) Conversely let of definition 3.31 and definition 3.41. that T IL order to show is a P -space we have by corollary 3 ~51 tnat we 1 need only show that /Ix, e // satisfies (3.32), (3 .. 44) and (3.51) • (3 .32): If /Ix, el/ f: l e E: E , then /Ix, e // ~ r- 1 for all r > 0 . //x,re/ b 1 or /Ix, e I= 0 for all e f E X for all Hence and by (6 .. 11) of definition 6 .11 = 8 • (3.51): a -/; O , Given x 6T and such that /a/ ~ /Ix, e // . e EE there exist a G:-A , Then -l /Ix, el/ ~ 1, x = ay, and (3.51) is satisfied. (3.44): Given a E: A , ~nd e E-. E we can pick r > /a/ . Hence if //x,re// ~ 1, then /Ix, e /I f: r-l and //(3x,e/l = r-l /fl= 1 for all / p; f: /a/ . Thus (3 .44) of definition 3 .41 is satisfied by /Ix, e /I . //y, el/~ /a/ It then follows by corollary 3.51 that normed w.r.t. D in such a manner that w.r.t., A, E valent to and the new pseudo-norm T may be pseudo- T is a P1 -space /Ix, el/* is equi- /Ix, e // • By (6.12) of definition 6.ll /Ix, el/ = /a/ /a- 1 / /Ix, el/ for all x ET and e EE ., If x ~ 8 , there exist /Ix, e // > 0 • This , follows from (6$11) . Hence /a/ /a-i/ = 1 for all a €: A , a ~ 0 • By lemma 6 .1 /a (31 = /a/ / (9 / for all a,~ f. A , and this completes the proof of the theorem. N.B. Hence we see by theorem 6.ll that every P - space is equivalent to an N2 -space 1 2 in the same manner that a P1 -space is equivalen t to an N1 -space.(See theorem 3 .5le) We might also note t hat since A is a division ring that 11 '3 l (2.33) of definition..;implies that given exists a VE. 'll such that We could consider here ~a space. for all /a/ f: / (3/ generalization of a_nOrmed li~~ar [81 • An N -spa ce w.r.t. A,_ is defined to be an N1 - sp a.c e w. r. t. 2 !\ . 'U. where A i s a v a.lued di vision ring such that . The type of definition that we might take is illustrated by theorem 1 and 2 of 1 aV c:: fU U E:. 'U. there la/' I pl = lo. f l for all a, ~ ·GA . Linear fu.nctioraon f 2 ~spa ces. 6.2. If f(x) is a ~ -linear function on a P -space T 2 w.r.t. • A, E to a P -space w.r.t. A', E' , then 2 / tp (a)/ 1 L /a/ ; and if f (x) is not the null ~ -linear function then it follows that p(a) 'satisfies (4.15)-€1:.1~) • Hence cp(a- 1 ) = (~(a) )- 1 , and one may easily verify that ~ (A) is a di vision subring of that (p(a) A' • is an isomorphic mapping 1 It is cle 2r of Hence we see that the existence of a function on T to T' subring isomorphic to requi res that A. on A ~ (A) C A'. ~-linear A' contain a division We shall now introduce a more · general concept of a linear function. li•li• Throughout the remainder of the chapte r T, T' and T'' will represent A'', E'' P2-spaces w.r.t. A, E; A', E' and respectively. We shall in view of theorem 6.11 consider our pseudo-norms to be the pseudo-norms with respect to which the spaces are also P1 -spaces. An additive function DEFINITION 6.21. T' f(x) on will be said to be linear 2 if given e' E: E' corresponds an e E: E //f(ax),e'//' f: /a/ such that /Ix, e /I f 1 for all T to there impJ.ies a EA. It follows immediately from theorem 4.11 that every 1 See 2 [14] , p. 9, definition 5. This definition could also have been given for P1 -spaces, though it does not seem to follow in Pl-spaces that the sum of two such linear functions is a inear function. 82 q>-linear function on T to to T• is a linear function on T' • An adaitLve function f(x) on T will be said to be bounded if gi ven e' E: E' there Definiti on 6.22. T, T correspond e E- E to such that /If (x) ,e •II f: /Ix, e II for all x ET • THEOREM 6.21. condition A necessary and sufficient/that an additive function QQ. f(x) T to T• be linear is that f(x) be bounded. The proof is quite similar to that of theorem 4.11, Proof: since we are considering our P -spaces to be pseudo-normed 2 in such a manner that they are also P1 -spaces. The proof will be left to the reader. It is then clear by virtue of theorem 6.21 that a linear function maps bounded sets into bounded sets. THEOREM 6.22. g_ set A necessary and sufficient condition that S C T be bounded is that given an open set containing the zero element there corresponds an such that Proof: U a ~A S c au. The sufficiency follows from theorem 4.12, since T is also a P1 -space. Necessity: Since [ /Ix, e /I ; x E- S] is bounded 83 for each S c aU(e) e E: E , we have that for some This can be shown in the follovdng manner. a EA • There exist /a 01 = /a/ I ~ I , it is then true that given r > 0 we can find an a E-A such that /a/ > r . Given • e C:.E there exists an a EA such that /Ix, e// f: /a/ for all x E:S • Hence by (6.12) /la-½, el/~ 1 , i.e., 1s C or s a e • row given an open se t u , /a/ ) 1 a - and since U(e) c:. U( ) N • e E-U , there exists by corollary 3.3l(a) a U(e)c. U , and the necessity of the conditon follows fro m the above. Let a P2-space F be the set of all linear functions on A, E to a P2-space w.r.t. T w.r.t. Then M(f , w) f EF and can be defined as in theorem 4.13 A' , E' . for alh w E- W , and the follo wing theore m can then be proved. THEOREM 6 .• 23., T to Tt m The set F of all linear functions Q.U be pseudo-normed with resp ect to the stronglx partially ordered set W in such g_ manner t ha t g_ ;E.2 -space with respect to A', W • M(f, w) F is is the pseudo- lli2.!:IQ.. It follows fro m theorem 6.12 and (6.13) of definition Proof: 6.12 that for any ' , ' E At fl , f 2 E: F t hat It can then be easily verified that F a 1 a 2 t 1 + a 2r 2 E F. is a v2-space with respect to a 1' f and hence A' . and Define rw = (re',S) M(f,rw) = rM(f,w) . (r > o) . It can then easily be verifie~using tne properties of //f(x),e'//' and the definition of M(f ,w) 4.13) , that F (see theorem At , R+W. is a Pn-space w.rat. t:, In this R+W = W. case ~ - differentials of functions with arguments and 6.3. values in P 2 -spaces. THEOREM 6~3ls §:. function A necessary and suff icient condition that F(x) on Sy c T to function is that_ there exists an ~ d t E:: Et there corresponds §:. Tt be g_ e E: E d '=- E ~ -approximation such that given such that //F(x) ,d'//' f: /Ix, el/ for all /Ix, di/ f: 1 • Proof: The sufficiency is clear since //~(a)F(x),dt //' f: /a.///x,e// =//ax,e// for all //x,d// f: 1 . Hence /lax,e/1 f: 1 and //x,d// L 1 implies that /I p(a)F(x) ,d'//' f: 1 . Let us assume that Then given exists x E. T , F(x) is a {j?-approximation function. /Ix, di/ f: 1 , a E A , a I- 0 , .and 1/y, el/ f: 1 y ET r > 0 and such that /a/ - /Ix, el/ £: r . Hence /I p(a- 1 )F(x),dt //' = 1 and , e E-E x =ay $ there 85 since //a- 1 x, e // =//y, e /I f: 1 /Ix, di/ f 1 . and Therefore, due to the fact that ip(a-l) = (<:p(a))-1,. //F(x),dt//t f: /p(a)/ ~ /a/ f://x, el/+ r for /Ix, d / / f: This holds for any 1 . r > 0 , and hence the theorem is proved. With the aid of theorem 6.31 we can sta te the following corollary. COROLLARY 6.31. A necessary and sufficient c~ndition that ~ function on §!l open set Sy c T , y <= Sy , to x =y is t hat there exists a be f (x) _p-differentiable at p-linear function ,. f(y;ox) of ox · the property that there exist an e ~E there corresponds g_ #F ( ox) for all 1 , d / / T to T' with and to each d' Ee Et for which 1 f:. /Ix, e II /Ix, di/ f: 1 , where F(ox) Then d C: E on Tt f(y;ox) THEOREM 6.32. = f(y + ox) - f(y) - f(y;ox) . is the If ~xn) ~-differential of f( x ) at x = y. is a sequence of elements of then g_ necessary and sufficient condition that lim xn = x /Ix n - x, el/ = 0 for each e E- E . is that lim Proof: The sufficiency is clear fro m coroll ary 3.3l(d). I'\~«> T, In order to prove the necessity we have by corollary 3.31(d) that given e f. E there exists an integer m = m(e) such 86 /Ix - xn., e // f: 1 ., Hence given r ) 0 we have that for n > m(r- 1 e) /Ix - ~, el/ f: r . T"nerefore lim ,, ~.,../Ix - xn, e II = 0 for all e f; E • that n ) m(e) implies that COROLLARY 6e32. lim xn ~ x e E: E A necessary and sufficient conditinnthat r >0 is that there exist an there corresponds an implies that m = m(e) such that n > m(e) /Ix - x n ., el/ !: r ., Follows from theorem 6., 32, since Proof: such that given DEFINITION 6.31. If for some element y ET f(a,y) r- 1 e E:. E is a function on and all TY for all defined O <. / a/ f: r (y) , then we shall say that lim f(a,y) a ➔ D if given et 6E9 = g(y) l there exists a positive real number 0 < I a/ L.. r ( e ' ) implies that II f (a, y) - g (y) , e '// t f:: 1 , such that N.B. 0 We see by corollary 6.32 that if lim f(a,y) = g(y) , a ➔ o then for any sequence ~an) E; A such that an :IO (n=l,:2, ... ) lim /an/ = 0 we have that and 11--,> o6 Hence 1 lim f(a ,y) = g(y) e n 2 lim f(a,y) has the same properties as the limit a ➔ O It can be shown that this is equivalent to lim //f(a,y) - g(y),e'//' = 0 /a/➔ o 2 See theorem 4.21. for all e' E- E' . 87 of a sequence of elements of Tf • ♦ THEOREM 6. 33 e to If F (x) Se c:. T , 8 E S , 8 .Qll fill open set satisfies the condition of theorem 6s3l, then Tt lim ( (~(a) )-lF(ax)) =8 for llil9h x <:- T .. Q; ➔ O Proof: By hypothesis there exists an given dtG-Et there correspond e E::. E d(dt)E-E such that such that #F(ax) 1 d tf t f: //ax,e// for all //x,d(d't)//f: 1 r 0 (dt) > 0 such that c For a given y E. T pick r 0 (dt) and ~ (r 0 (dt)) ~#y,d(rdt)f where r >#x, e# . Hence for O l /a/ ~ r (dt) 0 //ay,d(rdt)ff: l and , I #F(ay) ,rd' YI~ /lay ,el/ . Since 1 /( tp (a))- / = /(p(a-1)/ = /a/-1, 1/( cp (a))-½,(ay),d'//' < 1 for all T o '-/a/<. r 0 (d') . is an arbitrary element of and this completes the proof of the theorem. THEOREM 6.34 .. to y T' is If a functi(m f(x) p -differentiable at ~m fill g_pen se1 x = y , thgg lim ( ~(a))- 1 (i(y +~bx) a, o exists fillQ !§ ~qual to f(y;Dx). By hypothesis - f(y)) ) Sy C T 88 "F(ox) = f(y + ox) - f(y) - f(y;ox) is a (Q -approxiruation fur1ction., and hence by theorem 6 .. 31 a.nd theorem 6.33 1 lim ( ( f (a) )- F (aox)) = 8 0, ➔ o for a.11 ox E T • Now F(aox) = f(y + aox) - f(y) - ~ (a)f(y;ox) and for a "I 0 - f(y)) - f(y;ox) • This last equation combined with the above completes the proof of the theoreme 89 BIBLIOGRAPHY LlJ A. A. Albert, Modern Highe~ Algebra, Chicago (1937). (2] S. Banach, Surles functionelles lineaires, I, II, Studia Mathematica, V,,, l, 2 {1929-1930)., S. Banach, Theorie des pperations Lineaires, Warsaw, l932e • (4] D. H. Hyers, A @neralization of frechet's diff,erential, Bulletin of the .American Wrathematical Society, v. 45, 1939, Abstract #325Q (5] D .. H. Hyers., Loca.ll;y: bounded linear .to}ological spaces, Revista de Ciencias, Nr. 430(1941, pp. 555-573. (6] D. H.. Hyers, Pseudo-normed linear spaces and abelian groups., Duke Mathematical Journal, Vol. 5, No. 3, Sept., 1939 1 pp. 678-634~ J .. 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