Polynomials in Abstract Spaces Citation Highberg, Ivar Edmund (1936) Polynomials in Abstract Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/m4jf-f255. https://resolver.caltech.edu/CaltechTHESIS:10142024-212559795 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 1936 Record Number: CaltechTHESIS:10142024-212559795 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10142024-212559795 DOI: 10.7907/m4jf-f255 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16792 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 17 Oct 2024 19:24 Last Modified: 17 Oct 2024 19:25 Thesis Files PDF - Final Version See Usage Policy. 12MB Repository Staff Only: item control page

POLYNOMIALS IN .ABSrRACT SPACES Thesis by Ivar E. High.berg In Partial Fulfillment ot the Requirements for the Degree ot Doctor ot Philosophy Pasadena, calitornia 1936 INrRODUOI'ION This thesis really had its inception in some work of Fr6chet. In 1909, Fr6chet, (l}, gave a f'unctional definition ot a polynamial, where the polynomial is a real, continuous tunction of a real variable. His definition was essentially: t(x} tinuous, and it :tor some integer n, is a polynomial, if it is con,_nH u .,.(x) "' = o. He then show- ed that such a f'unction could be represented in the usual form. Gateaux (1), next gave a definition ot polynomials tor f'unctionals. His definition was that t(x) is a polynomial if, t( µ- x + J,..y) where and y. ?,. ,µ are complex numbers, and kJ (x,y) is a tunctional ot x In 1929, Fr6chet, (2), extended his definition of a polynomial to a class of abstract spaces called ttespaees algibrophilestt, essentially linear spaces with a topology. He practically showed tll,e equivalence ot his definition and the Gateaux definition, and did in tact in the case ot f'unctions on the real numbers to the abs,ract space. He, however, did not consider "espaces alg6brophiles" in which multiplication by complex numbers is allowed. R.s. Martin, in his 1932 thesis, considered extensively the properties of polynomials from the Gateaux standpoint in a complex Banach space. Some ot this ~as been published. See Michal and Martin (1) , and Michal and Clifford (1) • Michal , (1) , has also considered the detini tion o:f' polynomials by means o:f' polars. Martin conjectured that if t(x}; besides satisfying 11 Fr6chet •s conditions, were also Fri!ichet ditterentiable at x f(:x:) = o, then would be a polynomial in the Gateaux sense. That this is not enough is shown by the example :f(z) = z 2 , which possesses a Frechet diti'erential at the origin. To make Frechet•s definition equivalent to Gateaux• definition, it has been tound sutticient to add the requirement of Gateaux diftere~tiability to Fr~chet 's conditions. See Chapter II. An algebraic condition has also been found, see Chapter III. It was thought profitable to investigate the naiure of a function which satisfies Friehet•s corulitions in a "complex espace algebrophile", and this led to the definition of pseudo-polynomials and the consideration of their properties. :Pseudo-polynomials are the :!'unctions with which this thesis is mainly concerned, and in order to develop the theory, the work is arranged in the following manner. Chapter I contains the postulates tor the space, and as much of the theory as is needed in the sequel. For a more extended account, see Frlichet (2). Chapter II develops the theory of polynomials in both the Gateaux and Frechet sense, and consists mainly of rewriting theorems from either i1art1n or Fr6chet. Section 4, in which the equivalence ot the definitions tor complex spaces is proved, is, however, new. In Chapter III pseudo-polynomials are defined from what might be termed a generalized Gateaux sense. The theory is developed in much the same manner as polynomial theory. For every theorem on polynomials, there is a corresponding theor~ for pseudo-polynomials, with one or two exceptions. The main result of the chapter is section 3, the equivalence theorems. Chapter IV is an application of the theory to the problem ot generating a norm by means ot a Hemi tian bilinear form, and Chapter Vis an example concerning the relation between the modulus of a polynomial and its polar. Appendix A contains a theorem needed in Chapter III, and Appendix B contains a simpler derivation of the equivalence theorem of Chapter III, but which was not found until the other had been done. It seems as if most of the results proved in this thesis could be established tor a linear topological space. It was thought best, however, to retain the "espace a.J.gebrophile" at tbis time. Modular properties of polynomials are not discussed, and the theory of polars has been barely touched. There is more variety ot possibility in polars of pseudo-polynomials. or For example, Ji ?i may have the polar 11, ~2 ~. 11 2 • In general the notation is that of Martin's thesis, and we do not give long explanations which may be found there. The functional notation is that ot E.H. Moore, and we use some point-set theoretical .p.otation also. The numbers atter an author's name refer to the bibliography at the encl. In conclusion I wish to express my appreciation to Professor A•.D. Michal, who first suggested to me the problem ot polynomials in complex spaces, £or his criticisms and suggestions. TABLE OF CONI'ENI'S Page Introduction 1 Chapter I The Espace Algebrophile. 1 1. The Postulates 1 2. Some Theorems on Continuity 4 3. The Gateaux Ditterential 5 4. Examples or Espaces Algebrophiles 5 Chapter II Polynomials in E. l. 7 Polynomials on C to E(C) 7 Polynomials on E , 8 to E 2. Ditterences 11 A Second Definition of E-polynomials and the 13 Equivalence of the Definitions Chapter III Pseudo~polynomials. 1. 3. 18 C-Polynomials 18 E-PolynamiaJ.s 21 A Second Definition of Pseudo-polynomials and the Equivalence of the Definitions 26 Chapter IV Condition that a Banach Space be a Hilbert Space. 35 Chapter V 39 An Example of a Modular Property. Append.ix A Appendix B 43 48 References to the Literature 51 CHAP.PER I THE l!Sl?ACE ALGEBROPHILE In this chapter we give the postulates tor an "espace algebrophile" and develop as much ot the general theory as is used tb.i'ougbout the body of the work. 91 The Postulates A complex "espace algtbrophile" E(C), is a system which consists ot:A set e , ot elements of aey nature whatever, and which we may call abstract points; Families {vi(} , of sets, Vx , ot abstract points associated with ea.ch abstract point :x ; {vx] constituti11g the famUy ot "neigh- borhoods" , Vx , ot x ; Operations concerning the abstract points, or the abstract poi;its and the complex number system C, and denoted by the symbols, " +", " •", " 11 :x ll " ; A system ot relations between the set borhoods C:, , the sets of neigh- { vx} , and the operations denoted abffe. In the f'ollOWi;ng we sWl assume that equality has the prop- erties o:t an equivalence relation. I The Fir~t Group ot Conditions.. 2 = y +X • (x + y) + z = X + (y + z} • 2. 3. 4. + y X a £. C If , and X t:. E 5. a • (x + y) &. (a + b} •x 7. a • (b •x} a. l •x 9. , then (a •x) €. E • = a •x + a ,y • = a •x + b •X • = ab ,x. - X• If x €. E , then 11 x 11 1s a real number ~ o. 10. There eJCists one and only one element O in E, such that 11 0 11 u. 1t a •x 1J =O• = la l I/ X II • tor x + -1 y , and I will now state some We write x - y definitions which ere needed tor the second group of conditions. Abstract Line. The set of points z , where Z = X + t • (y - X) t lC , will be called the abnraot line -zy. It is obviou.s that the points x and y 0 lie on the line 0 • That the line xy coincides with the line yx , 'JJJlf£Y be seen by aettiDg t = l - t '. Tranalation. A transformation ot E into itaelt by an equation ot tlle form, will be called a translation. Homothetie. A transformation of E into 1tselt by an equation of the form, where x 0 is a fixed point in E, and a is a fixed point in C , 3 will be called a homothetic transformation. Continuity. It z = t(x,y) and z 0 = t(x ,y (x ,y ) , 0 0 0 ) 0 is a function defined on E,E2 te E3 , then z = t(x,y) , is said to be continuous at it tor every neighborhood V2 of z • borhood Vx ot x , 0 there exists a neigh, O such that when 0 and y e V~ 0 , and a neighborhood V~ ot y 0 x E. V.,_ 0 O , then z i V 2 • O This definition of continuity has bean phrased in terms of two variables tor convenience. For tunctions of one variable, we drop one variable f'rom the definition. Point ot Accumulation. A point x will be called a point ot aecunmlation of "8et U, it each neighborhood of x contains at least one point of U (different f'rom x ) • We now proceed with the second group ot conditions. II The Second Group ot Conditions. 12. It U.x and Vx are aw two neighborhoods ot x, then there exists a neighborhood Wx of x, such that Wx C Ux• Vx • 13. The logical product of all the neighborhoods ot a point x is the point x alone. 14. The transtorma.tion (x,y) at all points z = x +y (x,y) is coiitinuous in the pair of E. 15. Homothetic transtorma.tions are continuous at all points of E. 16. The points of accumulation of an abstract line lie on the line. l'l. A necessary and sufticient condition that a point x 0 of an abstract line be a point of accunmlation of ¥et of points U on the line, is that the lower bound of ux - :x: 11 0 be zero as x ranges over U (remaining distinct from x 0 ) . 4 92 soma Theorems on Continuity Theorem l. z =:t(y) and y = g(x) are functions defined in If "espaces algebrophiles", and continuous at ' Theorem 2. x =x 0 , is continuous at y = y 0 then z = f(g(x)) z = f(x,y) It z , and y is continuous at x =x is continuous in the pair (x,y) at is O • (x ,y 0 0 = h(to ) , =g(t) ' y =h(t) are continuous at X O= g(to ) , then z = t(g(t} ,h(t)) is cop,tinuous at t = t 'Yo Theorem 3. A linear combination is continuous and X O it g(u) and h(u) • z = e. •g(u} + b ·h(u) are continuous. In the above three theorems I have not mentioned the domains of definition in order to avoid prolixity. It is clear, however, that the domains may be situated in the same or different spaces. Theorem l follows from the definition of continuity. 12 in addition. Theorem 2 requires postulate The proof of theorem 3 requires postulates 15 and 14, and theorem 1. Theorem .4. If f( ;;\ ) is a continuous complex function of a complex varial>le, and u 0 is a fixed point in E , then y = f( .t\)•u 0 is a continuous function of A • Proof': Let yo =f( ?i o ) • U 0 , and let Vu <> Let us assume that the theorem is false. one sequence i\ 11 ~ i\ 0 , be a neighborhood of y 0 such that 0 • Then there must exist at least y"1 = t {71 n ) • u o is not in Vu tor .Jo any n • However, and hence y 0 is a point of accumulation of the set {Yn} . We thus arrive at a contradiction with the hypothesis and hence the theorem is true. ), 5 Theorem 5. If u i are fixed points in E , and f ~ Ci\ ) are continuous functions of a complex variable, then, (l) A. is a continuous tunction ot Proof: Use theorems 3 and 4, and mathematical iD.duction. 93 The Gateaux Differential Definition 1. Let t(x) be defined on an "espace algibrophile" E, , to a like space E 2 • Then f(x) will be said to possess a Gateaux differential at it tor aDY" z in E , , :x: 0 , I( 1i m + t. 2) XO - I (XO ) exists, indepen4ent of the way in which differential by t(x 0 ;z) • Theorem 6. F( t\) Let = t( ~) ,u 0 , where C having a derivative everywhere. ev~here. Proof: Since t E. .G t t➔ 0 t ~ t( 7') Then F ( l\) O• We denote the Gateaux is a function on C to is Gateaux differentiable r c~ + tpJ - r t1i J t lim t( A + t u ) t➔ o a continuous function or = t ' ( 71 ) , and since g{f.1 } •~ :t( 21 l - rtjl 0 is µ , we concl~de that the Gateaux differential = Ji f ('l\) •u 1 exists, and F( " ; f' ) Theorem 7. A function of the to:rm (1) is Gateaux differentiable 11" all the are derivable. t i (A) Proof: 0 • This is clear :trom theorems 6 and 6. 6 94 Examples of Espaces Alg6brophiles It is obvious that the complex number system C, is itself an instance of an "espace alg6brophile". We may take for the family of neighborhoods attached to the point z , the interiors of circles having z as a center and rational radii. A complex Banach space is also an instance, with spheres about a point as neighborhoods. formal proof and for other examples see Frecbet (2). For a CHAPrER II POLYNOMIALS m E The object ot this chapter is to give a definition ot polynomials• ley down some ot their tundamental properties, and to consider the equivalence ot two different definitions which have been given. The results ot this chapter appear in a paper which has not yet been published, Highberg, (1). Ql Polynomials on C to E (C) We begin by considering tunctions on C to E(C} • and the natural definition ot a polynomial on C to E(O) is as tollows. Definition l. C to E(C) • which can be A function p(l\ ) defined on expressed in the form, valid tor all (1) == where the ui Uo + /\, U , A , + • • • + n /1 •U n , are fixed elements in E , will be called a C-polynomial. The index of the non-vanishing u i with highest index will be called the degree ot tlle polynomial. A polynomial of degree O is a constant. A polynomial which is io.entically zero may be regarded as having any degree wllatever. Theorem l. Proof: A C-polynomial is continuous. This is a special case ot I, 92, theorem 5, with t i ( A) . • ,"L 1 8 Theorem 2. a C-polynomials ot degree A linear combination of 0-polynomial ot degree Theorem 3. pol.ynom:1al on C to C-polynomial ot degree m, and ot degree n, then C is ~ m. is a It p( A) ~ m , g(?\)•p( i\) g( 11 ) is a is a 0-poly- nomial ot degree m + n. Theorem 4. is a If p( i\) C-polynomial ot degree ui = vi q( 11 ) , where ui m, and C-polynomial ot degree q( ?I ) is a = q( i\) ; then m = n and n •, · and p( ,\) and v , are the coetticients in p( 7I ) and C-polynomial in the form (l) respectively. The representation ot a Theorem 5. is unique. The proofs of theorems 2-5 have been omitted, for the details are no different from the method in Martin's thesis, Martin (1), PP• 19-21 • Theorem 6. A C-pol.ynomial is Gateaux ditterentiable everywhere. This is a special case ot Proof: I, Q3, theorem 7. 92 Polynomials on E , to E 2 In this section we discuss functions defined on one "espace alg6brophile E , , to a like space Definition 2. Let p(x} E2 • be a function on E , to E 2. • Then p(x) is a C-polynomial in will bo said to be an E-polynomial, it l O 2° p(x) is continuous, tor every pair (x,y) , It Will be said to be of degree p(x + 11-y) ~ , it tor some (x,y) , p (x + °A·Y) 1' • is 9 a C-pol.yllOlllial in A ot degree n , and tor all (x ,Y) is a C-poly- nomial in ?i ot degree ~ n • An E-polynomial. can thei-etore be represented in the torm, (2) p(x + /\•Yl whei-e the k i (x,y) 2. are tunctions on E, to represented in the form (2), the f'Unctions associated functions to Theorem 7. When is p(x) are called the k i (x,y) p(x). ~ n A linear combination of E-polynomials ot degree an E-polynomial ot degree Proof: E2 • , is ~ n. This follows immediately from I, ~2, theorem 3, and theorem 2 above. Theorem a. It then p ( c •X + x O ) Proof: First, = is an E-polynamial in x p(c ,x + :x: 0 ) theorems 3 and l. c fc , n • by I, ¢ 2, n + t\· k , (C •X + x o,c .y) + ••• + A• k n (C •X + x o,c .y) p(x) that is, f ': k ,.,(x,y) 1s an associated ot degree r , is h8lll0geneous in y It p(x) ~ n • is an E-polynomial, and k r (x,y) tunction to it, then k ,Jx,y) Theorem 10. ~ ot degree is a continuous tunction of x and this is a C-polynomial in 71 ot degree It n , and Secondly, using (2), k 0 (c •x + x II ,c •y) Thewem 9. ~ is an E-pol.ynomial ot degree p(x) • is an E-polynan:1,al ot deg,:-ee ~ n , and k ,Jx,y) 10 is an associated tunction to it, then tor fixed x , E-polynamial in y ot degree ~ n E-polynamial in x ot degree ~ Theorem 11. If h(x) k r (x,y} , and tor :ti:x:ed y , is an k r (x ,y} is an n • is an E-polynomial ot degree n , and it it is homogeneous ot degree :r, then n = r . * Theorem 12. k r (x,y) If h(x) is an E-polynomial homogeneo~ ot degree m , and is an associated function to it, then k r (x,y) in x ot degree m - is homogeneous r , that is, k ,J f'• X,y) = f1 m-r" • k,,, {x,y ) • The proofs o-r the last tour theorems depend only on the linearity ot the space, the definition, and theorems 7 and 8 above. Similar theorems -rray be found in Martin (l} , pp.28-31. By setting x = o , .t\ = 1 , h ,..(y) = k ,Jo,y) in (2), we get the theorem, Theorem 13. If p(x) is an E-polynomial ot degree n, it can be uniquely represented as a sum ot homogeneous polynomials in the form, (3} p(x) Canonical Representation h 0 (x) + h I (x) + • • •+ h r, (x} • The representation ot an E-polynomial in the torm (3), will be called the canonical representation. ~ote: A function which is homogeneous is not necessarily a polynomial. For example, in a real Banach space, II t •x II 2. t 2 II x II 2 , but If x 11 2 is not a polynomial. It in fact it were a polynomial, the norm would be of Hilbert type, that is, generated by a quadratic torm. = ll 93 Differences Let F(x) be a function defined on one "espace alg~brophile" E, , to a like space E 2 • We define successive differences inductively. The first difference is written I fl F(x) F(x + u) - F(x) , ~ and the n~ difference with independent increments is defined as /J. n F(x} • 11,, • • •, u,., u,~ ~-,~~O} means In particular, [~,~~,;~x ) Ix= 0 • It can easily be proved by induction that the nth difference of a :f'unction is synnnetric --,,. in tp.e increments. It we apply this to a t'unction of the :form (1), we see that so that we may assert, Theorem 14. It p{x) is a C-polynomial of degree n , then is a C-polynomial ot degree ~ n - l , and tor some choice ot /J.' p( ~) µ f is ot degree exactly n - l . Proo:r: This is evident from the equation above. Theorem 15. p( /\) if"+.1 /ln ·• • -•,f'n-,., Proof: p( ,\ ) = is a c-polynomial ot degree n, then O , and tor some choice o:r the increments, [J.n p(i\) =/=- o . f,,' ' ',f,, Use theorem 14 successively. Theorem 16. t/ p(x) u It If p(x) is a homogeneous E-polynomial of degree n, then is an E-polynomial of' degree of degree exactly n - l . ~ n - 1 , and for some u is 12 Proof: Using the representation (2} and theorem 12, ~ = O , Putting ment. we see that we set k i (O,y) t' : i\ • is independent of the first argu- f =- O and ;\ = 1 , we get p(y) Further, setting From theorem 12, k n(x,y) = O if l , replace :x: ti I p(X} = k /') (x,y) • i i n, so if in the equation above, by u and y + k 1 (u,x) + by x , we get . .. • lJ By theorem 9 this is a sum of polynomials ot degree ~ and hence is of degree in x , To show the second part of the theorem choose u == i\, x • = [(l + 71 t 1].p(:x:} (/ p(x) p(X + /l• X) : Equating the coetticients ot n , p(x) p(x) tor which p(x) - is ot degree p(:x: , } =I= O • degree exactly n - 1. Theorem 17. p{x) It is an E-polynomial in = k n-1 (x,x) • n, there is at least one value of x, say x, , Choose u =x , • Then k is an E-pGlynamial of degree x n - , (x , ,x) :/=. O 1 /J p{x) x, , is of I n , then .1 p (x} u ~ n - 1 , and for some choice of p(x} in the canonical form (3} and of degree is an E-polynomial of degree n - 1. Use the representation of then employ theorem 16. Then A on both sides ot the equation we get, and hence is of degree exactly n - l , and from this, Proof: n - 1 n - l • i\..J(' Since ~ u 13 Theorem 18. uA n+t p(..-) .... 11,, • • ·, 11,, .. , Proof: p(:x:) It is an E-polynomial of degree n, then = O· , and tor some choice of the increments, 11, /j,', ·,pu,,x rd--. O n ( ) Use theorem 17 successively. Theorem 19. An E-polynomial is Gateaux dif':ferentiable. Proof: Using the representation {2) we have, p(x + 71. y) p(x) - Dividing by ?\ and letting 71 tend to zero, it is evident that the limit exists, and that p(:x:;y) 94 = k ' (:x:,y) • A Second Definition ot E-Polynomials and the Equivalence ot the Definitions In this section we shall give a. definition of polynomials, whicll tor real "espaces alg§brophilesn is due to Frechet. spaces, conditio~ 3° o:r definition 2 ' is unnecessary. In th.a real The equivalence theorems for complex "espaces algebrophiles" are new results. Definition 2•. Let p(x) p(:x:) be a function on an E, to an E 2 • Then will be said to be an E-polynomial, if' 1° p (:x:) is continuous, for some integer n, /J. n+ I p (:X:) - 0 11, ' • • •t 11 n+1 3° p(x} possesses a Gateaux differential everywhere. * It will be said to be ot degree n, if /j n p(x) u,,···,un $ O , for some *Note: The condition of Gateaux diff'erentiability '1!13.Y be replaced by a purely algebraic condition. See Chapter III. 14 This definition includes the case of polynomials defined on C to E, and in discussing the equivalence of definitions 2 and 2', it is convenient to consider the C-polynomials first. Theorem 20. In definition 2' let E, be C. Then definition 2' and definition 1 are equivalent. Proof': Tllat a function which is a C-polynomial of degree n by defin- ition 1, is a C-polynomial ot degree n by definition 2', is clear from theorems l, 6, and 15. To show the converse we must show that a function on C to E2 , which satisfies conditions 1° , 2° , 3° above, has the form (l). We make the proof by induction. Case I. In this case, t( t\ ) = u0 tl t(11 ) f,,f'I. case II. Setting , u0 a constant, and is of' form (1). =o • 71 = O in this equation we have, (4) Let us define ¢ (f-1) = :t (µ ) - f (O) • Clearly ¢ (µ ) is continuous and Gateaux differentiable everywhere. Also, using (4), (5) • Then by a well known limiting process, t Hence if ,\ :: ,\ + i ~2 , real. 15 "+,; q>(I) 2 71 - ;\ ¢ (i) zr , + ( 6) ' where ~ is the complex conjugate ot Since i\ • ¢ ( ;\) is Gateaux differentiable, and the first term on the right of (6) is likewise, we conclude that since ~ ~- u 2 is Gateaux ditterentiable. This is a contradiction does not possess a derivative, and hence u 2 must be zero. We have therefore proved that f(O} + A•U I t and is therefore of the form (1). Case III. f( ~) :: 0 lln+I • /J, ,' • • ,fn<·I Setting UM,= f I , ~-~ [t( ?I + f-' ) - t( ?I }] f',' 'p,., t( A + p } - t( ?i } , considered as a function of = 0 • Now, 7\ , is continuous and Gateaux differentiable everywhere.* Moreover, it f( A} is of degree n by definition 2', then f( 1t + _µ ) - tL~.} is ot degree n - 1 by definition 2'• Hence under the induction hypothesis we shall suppose that 1. f( ~ +f ) - t( ~} is a C-polynomial of degree n - 1 by definition Let us set, (7) Then 11' ( ~ , µ ) it}s symmetric in is a C-polynomial in /\ of degree 11 ,}-J n - 1 , and since , it is also a C-polynomial in f' of degree n - 1 • Note: It is essential that f( A) be Gateaux differentiable everywhere, for otherwise f( A + µ } - f( A) may not be differentiable anywhere. _2 Differentiability at one point is not sufficient. For example ~(~) = ~ is differentiable at ~ = O , but nowhere else. 16 Then by a purely manipulative process, the details ot which may be found in Fr~chet (2), pp.86-88, we tind (8) 1/1'{ t\ , µ > - g( A + f > - g( 'A ) - g{ µ > where and where the UJ are :fixed points in E • We set Then, using (8), Now t{ ,\) is continuous and Gateaux differentiable by hypothesis, and g( A) is likewise since it is a C-polynomial by definition 1. Hence ¢ (71) by Case II, is continuous and Gateaux ditterentiable, and therefore ¢ ( 71 ) = A·¢ (1} • Hence, and this is or the torm {l}. To show that the degrees are the same, 7t , we observe that the left hand side ot (8) while the right hand side is of degree r-1 • Hence r = n , and the is of degree n-1 in equivalence ot the two definitions is completely established. The equivalence theorem tor E-polynomials is now readily established, and we shall give it as, Theorem 21. Proot: Definitions 2 and 2' are equivalent. That a function which is an E-polynomial of degree n by definition 2, is an E-polynomial of degree n by definition 2' , is clear trom definition 2 , theorem 18, and theorem 19. To show the 17 converse, we will suppose that F(x) satisfies the conditions of definition 2' and is of degt"ee n. Then t( ~) continuous in 1 A , and is Gateaux ditterentiable everywhere. Moreover~ _ 0 • Hence by theorem 20, t(?I ) in ?I ot degree ~ n • Therefore, /1 1+' t( ,\) /J, J • • •, f'n♦, ~ n = F(x + ~- y) is F(x) by definition 2. To show that F(x) is a C-polynomial is an E-polynomial of degree must be of degree exactly n, we use theorem 18, and with that we have proved that the definitions are equivalent• CHA.Pl'ER III PSEUDO POLYNOMIALS In order to preserve the equivalence ot the Frechet and Gateaux definitions ot polynomials in complex 0 espaces alg6brophiles", it was tound necessary to add the condition of Gateaux differentiability to Frachet's conditions. It is of some interest to examine the nature of the tunctions which arise when th di:rterentiability condition is suppressed. It is proved in this chapter that such functions are pseudo• polynomials, that is,?!- or E-polynomials. The plan ot this chapter is to develop the properties of pseudo-polynom1als which correspond to the properties of ordinary polynomials, and finally, to show that a f'unction satisfying Frechet's original conditions is a pseudo-polynomial. 91 Definition 1. A function 0-polynomials p(i\ ) defined on C to E , which can be expressed in the form, (1) = u n 00 + i\•u 10 + ~-u ,, + .• ' + i\, u no + i\ n-1- n i\,u n, + • • • + tl-u nri , where the u rs are fixed elements in E , will be called a C-polynomial. It will be said to be of degree n , it not all the u n j are O • A c~polynomial of degree O is a constant, and a C-polynomial which van- ishes identically may be regarded as having any degree whatever. Theorem l. A C-polynomial is continuous. 19 <. m , is a A linear combination ot C-polynomials of degree Theorem 2. ~ m • C-polynomial of degree tt p(;\ } is a C-polynomial o:f degree n, and g( i\ ) Theorem 3. C ot degree m, then g( i\) •p( A) C-polynomial on C to nomial of degree is a is a C-poly- m + n. The proof of the three above theorems is similar to the proofs for C-polynomials. The next theorem which should be proved is that the representation ot a 0-polynomial in the form (l} is unique. be able to solve tor the u rs To do this we must in terms of values of p ( t. } • do this it must be shown that we can choose ½(n + 1) (n + 2} ?I , say t. 00 'A, , 0 , / , A,o , 1\0 n ~10 n-1 _ n i\00 , ' ... ' i\oo ~o• , • . ' 11," \o' W•I . . Aoon ' n A," ... - does not vanish. values of A,, , • • • 'Ano ' • • • Anr, such that the determinant I , II oo , ?i "° , / , In order to • • • , /\ nn , n ~ nn That this can be done is proved in the appendix. It MJ we denote by A ,. 5 the normalized cofactors of the above determinant, (1) can be solved for the constants, (2) Using this result we are enabled to prove Theorem 4. If p ( A) and = p( i\ ) and Let are C-polynomials of degree q( A) , then m = n are the coefficients in Lemma 1. q ('A ) p( ?t} = ,\ p{ A} and I" - s and u ,, 5 q( ~) A• u 0 • Then = vrs , where m and n , u ,.. 5 and v ,.. 5 respectively. tJ.' p m ) 11 is of degree r + s - 1 20 in i\ , and £'.jl"+s r ! s ! I:K TTK Tl<, u • • • IuK,. IuK r • • • Ii1K,..,.s • v p(ti) µ,,•",/Jr+s 11 where the summation is extended over all permutations of the increments, 0 r + s r unbarred, ands barred. Proof: which is ot degree r + s - l are, { i\ ". 5 • ?i A . The two terms of highest degree in S•I p + • Repeating this process successively we get the result in the lemma. Theorem 5. It p( ?i.} is a C-polynomial of degree ~ a C-polynomial of degree n - l , and for some I n, then {j p( ?i) JJ is f1 is of degree n - 1 • Proof: This follows imlllediate1y from the lemma and the uniqueness theorem. Theorem 6. If p( ,\ } is a C-polynomial of degree and for some choice of n , then tJ.""t' p( A) - 0 , µ,~ ... , fun+1 f-', , • • • , fin }J, ,• • ',}'n Proof: (3) If p( A) is expressed in the form (1), then using lemma 1, p(Ji) /jn µ, '• ••, fn ,- - +···+nu • r·, ... Iu.u ,, nn • That this cannot vanish fer all choices of the increments i'ollows fl-om the :f'aet that (4) = and by the uniqueness theorem, if this vanishes identically, u n,, == 0 for all i , and the original polynomial could not have been of then~ degree. That the n + 1 61 difference vanishes is evident. 2l. 9 2 E-Polynomials In this section we discuss functions defined on one "espace aJ.gebrophile" E , , to a like spave E 2 • Definition 2. p (x} Let p(x) be a function on an E , to an E 2. Then • will be said to be an E-polynomial, if l e, p (:x:} i s conti nuous, 2° tor each pair x,y , p{x + ~• y) is a 0-polynomial in ii • It will be said to be of degree n, if for all x,y a C-polynomial of degree ~ n, , p(x + ~- y} is and tor some :x:,y is a C-polynomial of degree exactly n. An E-polynomial can therefore be represented in the form, (5} p(x + A• Y) = k 00 (x,y} + ~- k 10 (:x:,y) + ~- k 11 (x,y} + •• • + I\~ k no (x,y) + i\"·'~-k 111 (x,y) + • • • + ~~ k n,, (x,y) , where the functions k rs (x,y} are on E 1 to E 2 • When p(x) is represented in the form (5), the functions k r 5 (:x:,y) are called the associated functions to p(x}. Theorem 7. A linear combination of E-polynomials of degree ~ n is an E-polynomial of degree ·~ n ! Theorem a. I:f." p (x} p(c-x + x 0 ) is an E-polynomial of degree is an E-polynomial in x of degree ~ n , and c e: C , then ~ n • The proofs of these two theorems is similar to that of the corresponding theorems in Chapter II, and are therefore omitted. 22 Detinition 3. A tunction t(x) ot degree r will be said to be pseudo-homogeneous and s, if ( ) = fl r_s f •f X t ( f" X ) /-" C • Theorem 9. If p(x) is an E-polynomial of' degree n, then the associated functions k r 5 (x,y) are pseudo-homogeneous in y ot order r - s and s. Proot: 1"-S-S k •• (x, f• Y) k oo (x,y) + • • •+ ti + • • + (?i.p) By theorem 4 we oan equate coefficients ot ~ ~· k r 5 (X, }' Y} r-s - _.s ('J.p)k ,. (x,y) 5 + • ·' +• •• • J"-S _ S ~ in the two equations, thus getting, k ,.s (x, ru,y) Theorem.lo. If p(x} is an E-polynomial of' degree n, and k rs (x,y} ia an associated function, then tor tixed x, nomial in y of degree ot degree n • Proof: ~ ~ n , and for fixed y of Using the numbers can solve tor k ,..5 (x,y} k ,..5 (x,y} is an E-poly- is an E-polynomial in x 9l, and the method given there, we in (5) as, (6) • Theorems 7 and 8 now assure us of our assertion, since on the right we have the sum of' functions which are E-polynomials in either x Theorem ll. If h(x) is an E-polynomial of degree n, and if it is pseudo-homogeneous of degree Proof: or y. r and s, then n = r + s. Setting x =O in (5} we have, • 23 Hence by theorem 4, r + s ~ n • On the other hand, In order :for this to be a pseudo ...polynomial, we must have for any m , (7) Applying {7) for Corollary 1. then ~ s - j m = n , we get It p(x) r + s ~ • 0 n , and hence is an E...polynomial and p{ A• x) r + s =n • = ,'i\ n• p(x) , p(x) has the form, p(x + i\• y) Corollary 2. then It p(x) is an E-polynomial and p( A• x) - i\ ~- p{x) , p{x) has the form, + ,'i\ i\• k 21 (x,y) • These corollaries are immediate applications of (7). Theorem 12. degree k mJ (x,y} r If p(x) and s , and is an E~polynomial which is pseudo-homogeneous of k mJ {x,y) is an associated function, then is pseudo-homogeneo~s in x and is pseudo-homogeneous in y Proof: of degree o:rm.degree m - j r + j - m and s - j , and j • The latter part of the theorem has already been proved. the former we have, To prove 24 ,.m-J°,;;J Equating like powers Ot 11 11 in these two equations we obtain, k mJ. ( µ• x,y) If in equation (5) we replace h rs (x) x, ;\ , y by 0,1,x and set = k ,.. (O,:x:) , we get 5 Theorem 13. If p{x} is an E-polynomial of degree n, it can be unique- ly represented as a sum ot pseudo-homogeneous polynomials in the torm, Canonical Representation The representation ot an E-polynomial in the torm (8) will be termed canonical. Theorem 14. degree r If h(x) is an E.-polynomial which is pseudo-homogeneous ot 1 and s, then b. h(x) is an E-polynomial in x o:f' degree 1J ~ r + s - 1 , and tor some choice of u is an E-polynomial 01' degree exactly r + s - 1. Proof: Let (7) we have n = r + s , and let h(x) k MJ. (x,y) that k rnj (O~y) (5) replace be represented by (5). Using = O unless j = s • From theorem 12 we deduce = 0 for m < n , and k ns (x,y) = h(y} • x by u , and y ti' h(x) u by x , and set '/I = l • In equation '!hen, , and by theorem 7, this is an E..polynamial in x of degree ~ r + s - 1 • Using (7) we t'ind that the terms of highest degree in the above equation are 25 To show the second part of' the theorem we must show that u can be chosen so that the above expression is not identically zero. To do this, set u = 7i· x • Then, h ( X + ii. X) - h ( X) = r-+J- m _ S-J A Equating coeff'icients o:t k n-1 1 S- I (x,x) Hence it we choose x , since h(x} KmJ'(X,X) • A in these two equations we have, ,\ and - !l s •h(x} ' k 0 _ 1 s (x,x) I = r •h(x) • so that h(x, ) =I= O , which is clearly possible is of degree n, we get ( s + r} • h (x , ) =I= o • For this value of x , !J' h(x) is of degree exactly r + s - 1 • Theorem 15. is an E-polynomia1 of degree n, then If p(x) x, is an E-polynomial in x of degree ~ n - I 6 p(x) 1J 1, and for some choice o:t u is an E-polynomial of' degree eJQictly n - 1. Proof: The first part of the theorem follows by expressing p(x} in canonical form and then using theorem 14. To show the second part of the theorem we will take only the highest degree terms in the canonical representation, ti' p(x} + . • • + !!J u Now consider h nJ• (:x) • 11 I h no (x) + . , , + Li ' h It is pseudo-homogeneous of degree 1J n - j Hence by the last part of theorem 14, the highest degree terms in nn (x) • and ~ 'X I j • h n1 (x) 26 consist of, (n,j] ( k 11-,,J._, x,x ) + (.nj] k n-1, J, (x,x} (n - j + j) • h nJ. (x) • Adding these terms together, we get 1 + • • • + /J. p(x) n - ~ ,, 0 (x) + h n, (x) + • • + h nn (x~ X Now it p(x) is of degree n, there must be at least one value x , • for which and for that value of x, ~' p(x) is of degree exactly n - 1 and hence the theorem. Theorem 16. If p(x) is an E-polynomial or degree n, then and tor SODl8 u, , • , • , u ,, Preof: , !}. n p (x) u,,•·•,u,, Lln+i p(x) = =I=, 0 • Use theorem 15 success1vely. 93 A Second Det1nition of Pseudo-Polynomials and the Equivalence or the Definitions Definition 2•. Let p(x) be a tunction defined on an E, to an E2 • Then p (:x:) will be said to be an E.-polynomial if, 1° p (x} 2° for some intege;i:- n , is continuous, ~ n +, p (x} u,, ••• ' u,,., = 0 • It will be said to be of degree n, if for same choice of u, , · • • , u n fjn p(x) =f:. 0 • u,, ···, 11,, It is to be remarked that this definition is the same as the definition 2' of chapter II, except that the requirement of Gateaux differentiability has been dropped. This definition, as in chapter II, includes the case o:r C-polynomials, and in proving the equivalence of' the definitions it is convenient to consider that case first. o, 27 Theorem 17. In definition 2', let E = 6 , Then definition 2' and , definition 1 are eq'\liva1ent. Proof: That a function which is a C-polynomial of degree n by definition 1, is a C-polynomial ot degree n by definition 2', is clear from theox-ems 1 and 6. To prove the converse, we must show that a function on C to E i which satisfies conditions 1° and 2 ° of definition 2' has the form (1). We make the proof by induction. Case I Case II I ,.1 :t( ;.\ ) = o. µ In this case f'( ;.\ ) r::.2 f ' ( ? l ) = u a constant, and it is trivial. , 00 = o. µ,,)'2. In this ease, using equation ( 6} ot t( ,\) f:(O) II, 94 , we have, 71 •u 10 + Ji, u II + which is of: the form (1) • Case III T}ien /J.n [t( A + I-' ) y,,'",f',, t ( ?i ) ] = O • Now f( /I + JJ ) - f ( ? i ) is a continuous function of ~ , and moreover its nth difference vanishis ot degree n, then for some choice of the es identically. If t( ,t\) increments, the {n - l)th difference of f( A + f-l ) - f{ ?I) does not vanish. Hence under the induction hypothesis, we will assume that t ( ?i. + JJ ) - t ( ?i} is a C-polynomial of degree n - l and will show that this leads to the conclusion that by definition 1, f( ?i. ) is a o-poly- nomial of degree n by definition 1. Let us set (9) Since subtracting a constant cannot affect the degree of a C-polynomial, 28 1/1 { ?I , µ ) is a C-polynomial in ?I of degree exactly n - l • Moreover, since it is symmetric in /l , }-' it is also a a-polynomial in degree exactly n - 1 . f-' of By using (9) we may verity a result which is of considerable use later, namely, (10} 1'( 71 + JJ + J ) - :r{i\l - t(f } - :r( J ) - The expression on the left is completely symmetric in the three variables 'A , j-J , iJ • Since in f 1/f { A , f ) is a C-polynomial of degree in A and n - l , we may write it as, (11} u 0 J ,-, ) + A· u ,0 (f ) U oo (?I) + Using the numbers A; _/ , A111 ,j ot equations {ll) for u rs ( A} f-'• U 10 (,\ ) + ~-u {u } + • • • " ,- + P•u ,, c11 > + •• • 9l, we may solve the latter of as, If we calculate the value of 1/1 ( /\ , Am j } from the first equation, introduce this into u rs (~) , and introduce this in.to the latter of equations (11} , we get an expression for > (12) L, where the degree A P. n - 1 l/f ( ;\ , }-' ) ,,_qr .sA A Af p• 9,r,s are constant elements of E 2 • in 71 , 71 together, and of degree Hence it is of degree in the form, P, q,r,s ' This expression is of' n - l in I-' , p together. 2n - 2 at most in the four variables l\ , 7i , f , p . If we group together the terms which are of the same degree in the four variables, we may a.rra.mge it in the form, 29 where r t for ~ 2n - 2 , and Therefore, real. ,p ) ~ ( ;\ , ~ , f We may therefore solve (14} for as a linear combin- 1// (L ~ ,L µ } , where L • is an integer say. ation of terms of the form From this, using (9) and (10}, it is easy to show that ~ is symmetric in i\ , p , and ~ {~ t f', ~ + ly symmetric in i\ , f' , tJ • of Ifs { A , ~, f' , p } • p, iJ , iJ ) We shall now attempt to find the special form We write it in the following form, 5 • y-' . = L..-A ll•a. (15} S•J- J S•I _ 5.j L,A I J + \' L iJ I j + J 0 J + !/1s ( ~ , A, J) , p ) is complete- + S-1 • , , J r + [. ~S-JDJ ·d.J + , E. ~ l·f · I J J p_q,. A 1' µ ·9,,, 9,,. + + + + where 5.j 5.1 ' ~ s - 2 = s l u + v + w + :x: =s 1 ~ u,v,w,x ~ s - 3 Because of the symmetry of b J- = a J f J. = e .i , t p ,q,r ~ p + q + r ~Ut,~ , f' , P l in A, f c s.j = CJ· , j P,q,r = g q,r,p 1 u,v,w,x = 1 v,11,x, w , we have d 5.j k p,q,r ' ' • =d· J t = h q,r, p ' 30 ~ ( II t f , ~ + P, tJ , "J ) The expression completely symmetric in i\ , f-' , v , ~ ( JI , ~ , p , jJ ) + so we now consider 1t. s + [ . . s-, S-J-J . U U•O,; 0 J I + .s + + L, . LJ 5-JLJJ• 0 . I 0 J S-1 + + [ + [ P -'1 r J 5-J j I { ' ll I\ ,11 • 9 P,q , r + + • J A p · eJ l' L.., + ~ ,, IJ 9 U_,. • 9 I\ ,-- r ' . + L) . ( ~ + f) 5-J tJ J • C . q,r,,.,J.. J J is 31 -;,..5-J ~ J In equation (16) consider the coefficients of They are respectively 2a j and a j and ur•J iJJ • Because of the symmetry we must have • o:r = a J. Now consider the coefficients of 0 • and • The condition of equality yields, o:r (17} • On considering the complex conjugates of the above terms, we are led in a similar manner to, ..!.. (~) -d I 5 J d, (18} J • We next consider the coefficients of As-,·-~,· iJ J. where we have permuted i\ and (19) g' . i) , and where t ~ i ,j 1 • From this we get, . . • t ,J ,s-,-J In a similar fashion we p:rove, h. . (20) . . t,J,S-t-J . ( 5J I) .e S-1 , 1, j ,k ~ 1, we get where 1. . 1 1 J,K ' . 1 • ~' VJ p K D s-, -J - 1< If we consider the equality of the coefficients of and ·1/µJ iJ t< j1 s- ,' - ✓• - K . 5-t-J-K ( I +J),h . K . . . • I +J I S-1 -J - K I I( On substituting the value of h from (20), we get '(21) • We have now evaluated all the coefficients of ~ ( ?i , X, p , p } in terms of c, , d , , and the eJ • It remains to find its special form. First we will find all the terms which have the coefficient c, • 32 They are :from (17), s-1 -L. (22} 5 i""" f,,J . ( ~) As -J J f' J • C I 5 I [ u, t fl) - 71 - f • c, S S SJ • Next we will find the terms involving e , • They consist of the coei'fieients of g . I I h . - (5;'),e, . - I I s- I - t . s-,,J,1-J but do not involve any terms 1'rom (J), e, ' Since the subscripts on the l uvwx . h term nmst be at least one, we see that the h term contributes nothing. The terms in lf5 which involve e , , are then i\ J• / p. e, (23) • We shall now calculate the coefficients of e t , where 2 ~ t ~ s Such coefficients arise :from (19), (20), and (21), and are g. . J,t,5-t-J h . . S-t 1 J,t-J • The coefficient ot e t S· /•t ) + L. . I + J is then, from (15), (S- J , J -t t) A A f' • S-t-J • et + + - 2. 33 - t [ 5-t + A ( ~ + µ) - ~ S-t - f' S•t.1 I-' + [ Cll + µ) s-t - s-t 71 f - s-t] [ - S• t s-t [ _ (~ + p) i ( A + fl_t [( i\ + µ) S-t - i\ - f st] • • et • ei +pJ - - - t it - p t] • e-1: p t] ·et + -tfl s-t· et -t A t - 'A ·{ 24) • In a manner similar to that in which we got (22) and (23), we find that the coefficients o:r e s-, and d , are, (25) (26) • We finally get then, using equations (22} to (26), the result that where ~ can be expressed in the form, e 50 = 1 c, , e ~r = e " s g {71 } $ Then = [.o J II S-J - , e s.s =-kd, . Let J i1 • e SJ. 'f/5 { it , 1i , f , p ) and • = and using (13), (27) • Let H( ?l ) be defined by, H{ 7' ) = ~ + f( t\ ) and (27) it may easily be verified that, g{ i\ ) - H( ,\ + fl ) • Using (9) = H( ~) + H(,U ) Now from the definition of H( 7' } it is clear that it is a continuous function. Hence, using the result of Case II, we have, H(J, ) = /l· U IO + ~- U II • • 34 Substituting this result in the definition or H( A) , we get , an expression :f'or f( A) as a pseudo-polynomial by the first definition. The degree of l.f ( ;\ , µ ) is exactly n - 1 in A by our induction hypoth- esis, and the degree o:f' the right hand side of (27) is exactly r - 1. Hence r = n, and we have completely proved the equivalence of the two definitions ot C-polynomials. The equivalence theorem for E-polynomials is now readily established. We state it as Theorem 18. Proof: Definitions 2 and 2' are equivalent. That a function which is an E-polynomial ot degree n by definition 2, is an E-polynomial ot degree n by definition 2' is clear from theorem 16 and the definition. To show the converse, let ditions of definition 2'. Then t( /\ ) and 6"" :t{ ,\ ) by definition 2. == O • Hence t(i\ } p(x) = p(:x + ?\, y) satisfy the con- is continuous in 'A , is a C-polynomial ot degree ~ n That its degree is exactly n, is assured by theorem 16, and thereby the equivalence is completely established. Note: In definition 2' of chapter II, the condition of Gateaux ditterentiability nay be replaced by an algebraic one. If the con- dition is left out we get a pseudo-polynomial by the results of this chapter. Hence if in equation (2), we equate u rs = O , it s -:/:- O , we get an algebraic condition on p( A) equivalent to the differentiability condition. IV CHA.PrER CONDITION THAT A BANACH SPACE BE A Hll.BER.r SPACE This chapter concerns itself with the problem of generating an arbitrary norm in a vector space by means of a Hermitian bilinear form. The definition of a vector space will be assumed as will also the notion of a Hilbert space. For the former see 1Brtin {l}, and the latter, Stone (l), In a Hilbert space H, notions of continuity and distance are II f II H derived from a real valued function function of two variables , which is generated from a { ( f , g } ) , which has the following properties, {l} lo {(t,g}} ~ C 2 0 ({ a •t 3 0 (( t + g , h )) 4 0 (( f 5 0 { ( t , f )} Vle then define = a · ((f,g}} , g )) , g )) ~ - {{ t , h )) a EC + {( g , h ) ) {{ g t t )) (( t 0, and , t )) - 0 implies t ll 1' 111-i = (( t , r )) ~ , and prove that ll t /lH =0• has the usual properties of a norm, namely, that it is real valued, vanishes only tor the zero element of the space, is triangular, and ll a•x llH =/ al· ll x ll • In a Banach space B, notions of continuity and distance are derived from a postulated norm, Hx ll • Frechet (3) has :recently given criteria which ll x U in a Banach space must satisfy in order that there exist a i'un,ction (( x , y )) properties (l}, and for which in the Banach space, which enjoys the ll x II -= { ( x , x )} ~ • These conditions 36 have been simplified by J. von Neumann and P. Jordan (l}, and may be stated as, Theorem l. A necessary and sufficient condition that there exist a ( ( x , y )) function defined in a Banaeh space B , having properties ( ( x , x ) ) ~ = II x II (l} , and the f'urther property that for all x , is that, (2} + ll x - y ll 2 tor every pair of elements x,y in B. Furthermore, the function (( x, y }} is defined by, The proof of this theorem may be found in the reference above, and is omitted because we prefer to give our own. ~heorem l'• A necessary and sufficient condition that there exist a ( { x , y ) ) defined in a Banach space B , having properties (l) , function and the f'urther property that 11 x II = ( ( x , :x: ) ) t/ !Ix ll == o 2 (3) u,. 11, 1 u, Proof: if The necessity is ev,dent, for II x II ;: ( ( x , x )) ll x + A.• y /12. = so that II x II 2 ½. ll x ll V2 for all :x: , is that, • is a priori continuous, and , then (( x, x }) + A (( y, x )) + ~ (( x,y )) + ~~ (( y,y )) is a pseudo-polynomial of' degree 2, and hence ~ 3 II :x: II u,, v,, u, 2 =O • Now to prove the sufficiency. Suppose therefore (3) holds. Then by III, * The corresponding simpler theorem in real Hilbert space was pointed out by Prof. Michal in his seminar on abstract spaces, 1935. 37 93, theorem 17, ll x ll 2 is a pseudo-polynomial of degree 2. -i U"• x ll so that II X 11 2 2. 2 = i\111l x ll 2 k rs (x,y) k 11 (x,y) the function so that = k 0 ,,(x,x) k 11 (x,y} are complex numbers. has the properties (l), We shall show that Setting y = x, and A on both sides or (4), we obtain equating coefficients or 2 9 2, corollary to theorem 11, has the form, where the functions ll x U , is a pseudo-polynomial of degree 2, pseudo-homogeneous of degree l and 1, and hence by III, II x t ft.• Y II But - = k ,, (x,x) k 11,(x,x:) has properties 1 " and 5" - k 21 (:x:,x) or (1). Further, , because of the pseudo-homogeneity ( see III, § 2, theorem 12} , we have, k 11 ( f'•x,y} = f-' k 11 (x,y), so that it has property 2 ° . llx+~Yll (5) - Now, 2 ;\~ k 00 (y,x) + Ji k 10 (y,x) + ti k ,, {y,:x:) + k 21 {y,x) • Equating coefficients in (4) and (5}, we arrive at (6) I: oo (y ,x) = k u (x,y) , k 10 (y,x} = k 11 (x,y) Taking complex conjugates of both sides of equation (4) and equating coefficients, we get (7) k 21 (x,y) o 38 From the latter of equations (6) and (7), we find k 11 {x,y) (8) k ,, {y ,:x:} 0 We have thereby proved that property 4 ° is satisfied, and there remains only the additivity. t/- II A• x 112. :/=: 0 , and in fact is a constant. Now Calculating it we find that ll~·X + 1.1,11 2 = -A,; 1(00 (x, 11,) - 7i k,o (x, 1.1,) If we add the four equations above, since k 00 {x,y) y,weget ?\· {k 10 {x,u I + U2 ) + 'A• {:le ,, (x,u, + u , ) + {k 21 'tx,u , + u 2 ) is independent of - k ,o (:x:,u , ) - k 10 (x,u2. l} - k (x,u , ) - k (x,u, )} - k (:x:,u , } - k {:x:,u • 11 11 21 21 2 )} It this is to be constant, we must have in particular, k 11 (x, u , -+- u 2 ) (9) = k 11 (x,u ; l + k ,, (:x:,u 2 } Equation (9) together with (8) enables us to show that • k 11 (:x:,y} satisfies property 3 ° , and with that we have demonstrated the theorem. This proof might have been simplified right from the beginning, tor from (3) it is easy to show that (2) holdso written as, l::,, 3 In tact (2) can be 2 II OU = O , .which is a special case of (3}. We x, y,-!:I preferred, however, to bring in the polar from the expansion. CHA.PrER V AN EXAMPLE OF A MODULAR ffiOPERTY For the notation of this chapter see Martin (1), p. 450 1vmtin proved that if' h(x) is a homogeneous polynomial of' degree n in a Banach space, and if' its modulus is denoted by M, and the modulus of' its polar is denoted by MP , then the moduli satisfy the inequa.J.ity, M M • However, it was not known whether this bound could be reduced, that is, do there exist homogeneous polynomials, the modulus of' whose polar comes nn arbitrarily close to nl M • That the answer is affirmative in the case n = 2, is shown by the following example. We shall be concerned with C , functions defined in the interval a ~ t with, Of II X II = max Q ~ t 1,, b the space of continuous ~ b • This is a Banach space I x(t) I . Now consider the following function X, + f K(s) {x(s)} ds 2. b 2. H(x) {x(t )} a where K(s) is a continuous function. H(x) is a homogeneous poly- nomial of degree 2, for 1° it is continuous, 2° H(x + ~- y) is a polynomial in 71 , • As such it has a modulus M, satisfying the relation, II H(:x:) 11 ~ M ll x ll 2 • 41 .K 2 , where the integral, under this sequence of functions, is K {I K(s) I ds • 0 2. Since the integral. is added to {x(t )} M = , clearly the modulus of H 1 + - K is • 2 Now consider the expression, = x(t) y(t) / ~(s) x(s) y(s) + ds • 0 H 2 (x,y) is the polar of H(x) H , (x,x) = H(x) ·~ Then, t for it is bilinear, symmetric, and • To calculate MP , II H 2 (x ,Y} II we must get the least upper bound of while ll x II II y ll = 1 • to talce II x II = II y II = 1 • y(t} = 1 x n (t) = 1 x n(t} = n (½ - t) x n(t) = - 1 It can easily be proved that it is sufficient Much as in the first part , we choose 0 1 ~ 2• .1. + .!.n 2 ~ t - 1 + K • We have therefore given an example in which, < MP and MP < n 2M ~ 1 K, and as before, we i"ind MP J_ < t < .l.2 + ..!.n I 2 - ri t 1 t ~ In this case• the least upper bound of the integral is M , • • 42 obv r'ou.s It is alsoAthat we can make Mp M approach arbitrarily near to tor K can be increased by multiplication ot K(s) by a real number. That is, IT K I -; K 2 ~ 2. 1 as K-+ o0 2 , • APPENDIX A In order to solve the problems ot Chapter III, it is necessary to be able to solve the equation, = (1) n by means ot determinants. f ( i\ ) = 0 , then a ,,_j _J a oo + i\. a 10 + i\- a ,, + • • • + A• a no + • + A i\- a nJ. sj t In other words we wi13h to show that if = 0 , for s = 0, • • • ,n , j = 0, • • • ,s • In order to do this we must show that there exist complex numbers r i\, , · • · , A,,, = ½n(n + l) + j + l such that the determinant / , ii, ' D .a, , • • • ' i\,n ' • ii n-J - J • ' ' it, .... . . .. n n-J -J t ' Ar, Xr' . . . ' II,., ••• 'i1,. !l,. does not vaaish. We shall make a proof by induction. Therefore we assume that there exist values ?I, , • • • , 71 ,. such that we now wish to show that we can choose a value of determinant, which follows, does not vanish. /\ D * O , and , so that the 44 'l ~ 112 I /12 ' / • • • ' 'l t1 /12 I • • • ' ' 'l n-J7:iiJ /I /I l. n-j-/ _ j+I i\2 2 ' 'l fli Expanding this determinant by elements of the last row, we get {2) where b n,J+i = D '1' 0 • We wish to show that ¢ (/\ ) has at least one non-zero value, or in other words, that it cannot vanish identically. In order to prove this, let us make the supposition that ¢ (~) - O , and show that this leads to a contradiction. '/1. = x + iy ?i = JC - iy Let us set and b rs + ib"1"S • Then by substituting in (2), ¢, (x,y) where ¢, and ¢2 are real i"unctions ot the real variables x,y. Hence if ¢ (/\ } - 0 , then since ¢, (x,y} and ¢ 2 (x,y} ¢, (x,y) = O , ¢ (x,y) = O • Now 2 are polynomials in x,y which vanish identically, all the coefficients in the representation must vanish. In particular all terms whose total degree is m must vanish identically. Clearly, these terms arise only from the terms in (2) which are of total degree m, say 45 {3) • II Consider the term ,/\ m-.s - s ?I . = (X t / 1:J) m-5 . m-s -.S t\ • Replacing ?I :x: + iy , S ( X - I t:f) • In the above expression the coetticient of x rn-p (iy) f 'E. (-) (m;.s HJJ (J ·( 4 ) by d.f is • 13 =fl Or rewritipg, i\ m-s - s ,1 yields, m )' b ¼j f1l, L,f' X o m -f Y m i"' hm, xm-11(,' .µ mJ ~f' ,µ,J Y, fl r . (i'J f' p m. . b I • L-,J UJ o Since the coetf'icient ot r' x m-1-1 y f' sums when the coetticient of we may assert that if' + mJ is real in the first of the above x m-1-' y JJ in the second sum is inaginary, = 0 , then h m ( 'A , ~ ) =0 • 0 These equations have one solution b mj ' = b mJ ". = O prove that that is the unique solution. , and we wish to To do that, it is autticient to show that . det fl,J = o, • •·,m Now consider (1 + x) m- iJ / p ~j / r' (1 .. x) i) =I= O • This is equal to • • If we set x = -1 , we find 46 (5) == 0 - ' 2 ,,, , tor U == o, • • • .m - 1 tor ,J =m • Hence, m m o,o , • • • , Po,m ....... (6) • m m Pm.1,0 '• • 'Pm.1,m-1 This is evident on multiplying every other row in the determinant on the lett by -1, adding to the last row, and using (5). Now consider the determinant on the right ot (6). det Ip;;~ I now show that it is • I shall By writing out the expressions and using (4) it is easy to verity that m ('l) • Pµ,µ m Also, tram the definition, == 1 • P o,u determinant on the right of ( 6) , we replace Hence , if' in the P m-1 o,t..l by , and subtract the :f'irst row trom the second, we find that the elements of the second row become P m-• ,,&J Subtractina this row f'rom the third, -0 • and so on, we finalJ."1'1' reduce the determinant to ., f) , U - 0, • • • , m - 1 • (-2) ½m(rn+1) = O , for all m and diction since b n,J+I . ' , and theretore, b mj ,IJ Hence, = Hence in (3), I p m-' I /J =I=- O • *0 • j , which leads to a contra- Therefore we have proved, that :ror any 47 r i\, , • '- • , Ar , there exist numbers such that the determinant D does not vanish, and hence that the equation (l) can be solved for the coef:ficients a mj • APPENDIX B In a paper which has recently appeared in Studia !Jathematica, s. Mazur and w. Orlicz, (1), have made some contributions to the study of polynomials which are very fundamental and beauti:f'ul, and which can be used to simplify some of the proofs in this thesis. First we must have some definitions; these hold in any linear space. K-additivity: A function U : (x , , x 2. , • • • ,x K} will be said to be k- additive if it is completely ayIIJmetric and additive in each place. Rational homogeneity: A function U1< (x) will be said to be rationally homogeneous of degree k, if it can be expressed as U" (x) where U : (x 1, • • • ,xK} U ~ (x , , • • • ,xK) == U : (x, • • • ,x) is k-additive. In thismcase, is called the generating function of U K(:ic) • They are then able to prove the following fundamental theorem. Theorem and if li U(:x:} is defined on one linear space E, b. "•' U(x) _ to another, E 2 , 0 , (where the increments are all the same), then u, • • •,u U(:x:) is uniquely representable as the sum of rationally homogeneous tunctions in the form, U(x} + U , (::x:) + • • -' + Un (x) , 49 and where moreover, = • The proof of this theorem is rather complicated and the reader is referred to the Studia article for details. Let us now assume that A = 71, + i i\ 2 , A is a complex rational, that is, where i\ , ,\ are rational reals. Then it U1< (x) is rationally homogeneous of degree k, f. S L, s 0 K·S i\ ,\ ~ (A ~ ?j / ( \-,.X) K·S ( : ) l u: (~ ~) S K K )' \' \' ~ s L:m ,,~ m 711<-mflm(-i)K·~-Jt K•.S (,,.S)('.s{)('<t·SJ ft< Ull-( " K ,ix,,••••••,,1ix,) ~ '---y---" 5 If we now define H K m(x} • K·S as we have Using this result we see that U(x} U(x} where for A a rational complex number. is uniquely representable in the form + H 11 (x} + • • • + H no (x} + • • + H nn (x), 50 It the spaces are, besides being linear, also "espaces algebrophiles", then it is easily seen by taking a sequence complex rationals tending to U( t\, x) {i\} of A , that + ~ •H ,, (x) + ••• , where A is any complex number, provided ot course that U(x} continuous :tunction~ is e. REFERENCES TO THE LITERATURE Frechet, Maurice (1) Une Definition Fonctionelle de Polynomes. Nouvelles .Annal.es de Ma.thmnatiques, Series 4, Tome 9, 1909, p.145. (2) Les Polynomes Abstraits. J"ournal de Ma.thematiques, Series 9, Tome a, (3) pp. 71-92 (1929) Sur la definition axiomatiques d'une classe d'espaces vectoriels distancies applicable vectoriellement sur l'espace de Hilbert. Annals of Ma.thematics, Series II, Vol.36, PP• 70 ~i-718 (1935} Gateaux, R. (l} Sur Diverses Q,uestions du Calcul Fonctionnel. Bulletin de la Societe Ma.th~matique de France, Tome 50, pp.1-57, (1922) Highberg, I.E. (1) A Note on Abstract Polynomials in Complex Spaces. (To be published in the J"ournal de Ma.thenatiques) von Neumann, J". and P. J"orda.n (1) On Inner Products in Linear Metric Spaces. Annals of W .athematics, Series II, Vol.35, pp.'719-23 (1935) :Mazur, s. and W. Orlicz (1) Grundlegende Eigensohaften der Polynomischen Operationen studia Mathematica, Tome V, pp.50-68, (1934) Michal, A.D. and R.s. Martin (l} Some Expansions in Vector Space • .Journal. de Ma.th~tiques, Series 4, Tome 13, pp.69-91 (1934) ~achal, A.D. and A.H. Clittord (1) Fonctions Analytiques Implicites dans des Espaces Vectoriels Abstraits. Comptes Rendus, Tome 197, p.735 (1933). NJ.chal, A.D. (l) Notes on Abstract Vector Spaces. I Definition of Abstract Polynomials. (Unpublished) stone, M.H. (l) Linear Transformations in Hilbert Space and their Applications to Analysis. Colloquium Publications of the American Mathematical Society, No. 15, (1932)