Scholars' Mine
Doctoral Dissertations Student Theses and Dissertations
1972
Elementary length topologies constructed using pseudo-norms with values in Tikhohov semi-fields
Jackie Ray Hamm
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Recommended Citation Hamm, Jackie Ray, "Elementary length topologies constructed using pseudo-norms with values in Tikhohov semi-fields" (1972). Doctoral Dissertations. 189. https://scholarsmine.mst.edu/doctoral_dissertations/189
This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. ELD~ENTARY LENGTH TOPOLOGIES CONSTRUCTED USING PSEUDO-r~ORt~S
WITH VALUES IN TIKHONOV SEMI-FIELDS
by
JACKIE RAY HANt1, 1939-
A DISSERTATION Presented to the Faculty of the Graduate School of the
UNIVERSITY OF t1ISSOURI-ROLLA
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY in
t·1ATH EMAT I CS T2786 1972 57 pages c. I ,·, : / i i
ABSTRACT
Elementary length topologies defined on normed and pseudo-normed linear spaces are studied. It is shown that elementary length topologies constructed with different pseudo-norms are never equivalent. Elementary length topologies are constructed on certain topo logical spaces and some of their properties are investigated. It is shown that certain ••measuring devices 11 (i.e., norms, pseudo-norms, semi-norms, and pseudo-metrics) which take their values in Tikhonov sernifields may be used to construct elementary length topologies on any topological linear space. Relationships betv..Jeen two elementary length topologies generated with different measuring devices are considered. Let (X,t) be a topological linear space such that t is determined from a convex functional, p-norm or quasi-norm. The function q may be used to construct an elementary length topology on X. Let Xr denote X with this elementary length topology and let C(Xr,k) be the set of all bounded, continuous, complex valued functions defined on Xr. Some of the properties of C(Xr,k) which may be used in a quantum mechanical scattering analysis involving elementary length are considered. i i i
AC~JOWLEDGEME N TS
The author wishes to express his sincere appreciation to Dr. A. J. Penico and to Dr. Troy L. Hicks of the Department of Mathematics for their guidance and encouragement in the preparation of this dissertation. He is also indebted to Dr. 0. R. Plummer of the Department of Mathematics for his discussion and suggestions concerning the material given in Chapter VII of this work. The author also expresses his gratitude to his wife and family for their patience and understanding during these years of graduate study. iv
TABLE OF CONTENTS
Page
ABSTRACT • •.•••••••.•.••.••.•.••••••.•••••••••••.•..•...••••..•...•.• i i
ACKt·JOWLEDGEt"1ENTS •••••••••••••••••••••••••••••••. .•••.••••••..•••••• iii
TABLE OF COf~TENTS •••.••••..•••••••.••••.•.•.•••.•••.•••...... •••.•.. i v
I. Ir~TRODUCTIOr~ ..••...... ••••..•.••••..•.•..•.•...••....•..••. .. 1
II. REVIEW OF THE LITERATURE •••••••••••••••••••••••••••••••••••••• 3
III. CONVEX FUNCTIONALS •••••••••••••••••••••••••••••••••••••••••••• 6
IV. LOCALLY BOUNDED TOPOLOGICAL VECTOR SPACES •••••••••••••••••••• 12
V. LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES ••••••••••••••••••••• 22
A. LOCALLY CONVEX SPACES AliD TIKHONOV SEt·1I-FIELDS ••••••••••• 22
B. ELEt1ENTARY LENGTH TOPOLOGIES FOR LOCALLY CONVEX SPACES . .••••.••.....••.....•.••.•••..•••.•.....••. 24
VI. TOPOLOGICAL LINEAR SPACES •••••••••••••••••••••••••••••••••••• 38
VII. QUANTUM r~ECHANICS USING ELEr-1ENTARY LENGTH TO PO LOG I ES • •.•.•••.••••••••...•.•...•.•.•..••.•.••.••..•••••. 42
VI I I. SUMMARY, CONCLUSIONS AND FURTHER PROBL Et1S •••••••••••••••••••• 50
REF EREt~CES . ••••.••••..•••••.•••••....•.•.....•.. _•.....•.••.•..•.... 52
VITA ...... •...... 53 1
I. INTRODUCTION
The term "elementary length" is used to denote a fundamental physical constant A > 0 such that certain .. length measurments" are integer multiples of A. The existence of such a constant has been contemplated by Pythagoras, ~Jerner Heisenberg, Bertrand Russell, Poincare [1] and many others. George Gamow [2] has conjectured that the introduction of A into theoretical physics would initiate a "complete" understanding of physical phenomena. Although an adequate theory based on the existence of A has not yet emerged, some recent attempts have been made with some apparent success [3]. These atte~pts involve topologies which are derived from the assumption of the existence of A. Gudder [1] has shown that each norm on a linear space may be used to generate an elementary length topology. The topology of a normed linear space may be generated by many distinct (but possibly equiva lent) norms. It is therefore important to investigate the relationship of elementary length topologies which are generated ~'lith uifferent norms. It is shown, in fact, that distinct but equivalent norms will generate inequivalent elementary length topologies. Furthermore, the above questions involving norms are generalized to questions involving, in place of norms, other "positive" functions taking their values in Tikhonov semi-fields, and answers similar to those mentioned above are obtained. The problem of determining those linear spaces which allow the construction of an elementary length topology on them is considered. 2
When such a construction is possible some of the properties of the topologies are investigated. The set of all bounded, continuous, complex-valued functions defined on a space with an elementary length topology is important in attempts to set up a quantum mechanics involving elementa~ length. Some of the properties of this set and one of its self-maps are considered. 3
II. REVIEW OF THE LITERATURE
In 1966 a backward, large-angle scattering which could not be explained by the existing quantum electrodynamic theory was reported by Blumenthal, et al. [4]. Atkinson and Halpern [3] suggested that this unexplained scattering may be caused by a 11 topological potential 11 due to the existence of an elementary length. The elementary length topologies presented in [3] are introduced in an ad hoc fashion and it is explained that there may be many such 11 non-usual 11 topologies which involve an elementary length. In [1] Gudder gave definitions of 11 length 11 and 11 elementary length 11 and showed that the elementary length topologies introduced by Atkinson and Halpern can be derived with the aid of these concepts. It will be necessary to refer to the results obtained by Gudder throughout this work. The following is an outline of those results.
Let (X, II II ) be a normed 1i near space over the rea 1 or camp 1ex numbers. DEFINITION G1. A length on X is a function f : X~ {n\ : n = 0, 1, 2, ···}where\> 0 and
= n\, 2, .... (Ll) if II x II = n\ then f ( x) n = 0, 1, ' (L2) if llx111 ~ llx2 11 then f(x1) < f(x2); (L3) if II x II > 0 then f ( x + ~~~II) = f ( x) + A.
LEMt~A Gl. If IIYII >A then f{y) = f{y- ~) + \. DEFINITION G2. If r is a real number let [r] be the smallest integer which is not less than r, and let {r} be the smallest integer which is strictly greater than r. 4
THEOREM G1. A function f on X is a length if and only if f has one of the two forms:
(1) f(x) = A[llxll-aJ vJhere A> 0 and 0 < a < A;
( 2 ) f ( x ) = A [I I x11 -a J \'lh ere A > 0 and 0 < a < A •
THEOREf~ G2. Let A be a subset of (O, oo ) wi til 0 as a 1 imi t point.
If fA is a length for A E: A, then lim fA(x) = llxll for every x E: X. A-+ 0
DEFINITIOI~ G3. An elementary length on X is a function f : X -+ { nA : n = 0 , 1 , 2 , • • • } f o r A > 0 \'Jh i c h s a t i s f i e s ( L1 ) , ( L2 ) and
(L3') f(nx) ~ nf(x) for n = 0, 1, 2, • • • and every x E: X. THEOREt1 G3. A function f on X is an elementary length if and only if f(x) = A [ll~ll] for every x E X.
COROLLARY G1. An elementary length is a length . THEOREM G4. A length f is an elementary length if and only if it is sub-additive. COROLLARY G2. If f is an elementary length on X, then p(x,y) = f(x-y) is a metric on X. Gudder shows that the metric topology obtained from the metric p of corollary G2 is the discrete topology. He then explains that the discrete topology is physically unsatisfactory and obtains a non- discrete topology by proceeding as follows . If f is an elementary length on X it is clear that d(x,y) = if(x)- f(y) j is a pseudo-metric on X. 5
DEFINITior~ G4. For each real number r > 0 let
B(x,r) = {y : d(x,y) < r}. The set B(x, ~ ) will be called a simple ball and a simple ball will be denoted by B(x).
LE ~1t· 1A G2. Two simple balls are either disjoint or equal. LEMMA G3. Every ball is a union of a finite number of simple
balls.
THEORH·1 G5. The balls B(x,n~) are a base for a non-discrete
topology on X. Every (bounded) open set is a (finite) countable union of simple balls. If X r 0 then every simple ball has the form
j~x ) · - 0 1 B( x+llxll ,J- ,±, ±2, ••• .
Let Xc denote X with the norm topology and let Xr be X with the elementary length topology descriued in theorem G5. Let X+" T : X x X-+ Xc X Xr be defined by T(x,y) = (y, y-x) . Then the weakest topology on X x X for ~vhich T is continuous is the elementary length topology on X x X which was derived in [4]. 6
III. COf~VEX FUf~CTIOHALS
In this chapter elementary length topologies generated with the aid of norms, pseudo-norms, quasi-norms and Minkowski functionals will be considered. Let (Y,T) be a topological space and T : X 7 Y. If T is one-to-one and onto, and if t is the weakest topology on X for which T is continuous, then (X,t) and (Y ;r) are homeomorphic. Thus if t is the elementary length topology on X x X as defined in chapter II then (X x X, t) is homeomorphic to Xc x X r . In the results to be presented in this work the particular measuring device under considera- tion may be used to determine the topological space X . Thus the c specification of the topological space X determines the elementary r length topology ton X x X. This topology vJill be derived by carrying over Gudder's approach in the various cases. Let X be a linear space over the real or complex numbers. DEFINITION 3.1. A real-valued function p on X is a convex functional if it satisfies
1) p(x) ~ 0 for every x s X;
2) p(ax) = ap(x) for every a ~ 0; 3) p(x + y) :;; p(x) + p(y). DEFINITION 3.2. A real-valued function p on X is a pseudo-norm if it satisfies
1) p(x) ~ 0 for every x s X; 2) p(ax) = lalp(x) for every x s X and every scalar a; 3) p(x + y) :s p(x) + p(y). 7
DEFINITIOr~ 3.3. A real-valued function p on X is a quasi-norm if it satisfies
1) p(x) ~ 0 for every x E X;
2) if p(x) = 0 then x = 0;
3) p(ax) = lajp(x) for every x E X and every scalar a;
4) there is a k ~ 1 for which p(x + y) :s k(p(x) + p(y)) for every
X ,y E X. DEFINITION 3.4. A pseudo-metric on a set X is a mapping d : X x X ~ [O,oo) which satisfies
1) d(x,x) = 0;
2) d(x,y) = d(y,x);
3) d(x,y) < d(x,z) + d(z,y).
DEFINITION 3.5. A subset U of X is absorbing if for each x E X there is a real number r(x) > 0 such that x E aU for every scalar a such that lal ~ r(x).
DEFINITION 3.6. The r~inkowski functional of an absorbing set U is a mapping p : X ~ [O,oo) defined by p(x) = inf{a > 0 : x E a U}. It is shown in Taylor [5] that the Minkowski functional of a convex, absorbing set which contains 0 is a convex functional. THEOREM 3.1. All of the proofs in [1] remain valid if norms are replaced by convex functionals. The details of the proofs are essentially unchanged so they \-Jill not be repeated here. We note also that, with the exception of theorem G4 and corollary G2, the proofs in [1] remain valid if norms are replaced by the Minkowski functionals of absorbent sets containing 0 which are not convex. This allov.Js the construction of elementary 8 length topologies with the aid of any non-negative, positive homo- geneous function, and in particular, a quasi-norm. Two norms on a linear space X are equivalent if they generate the same topology. It seems natural to ask if equivalent norms generate the same elementary length topology. Before ansv.Jering this question, some results which are used repeatedly in this work will be given. If p is a non-negative, positive homogeneous function defined on
X then d(x,y) =A/ [p~x)J- [EflJ/ is a pseudo-metric on X which generates the elementary length topology determined from p. By theorem G5, a base for this topology consists of simple balls of the form B(x) ={y : d(x,y) PROOF. If z s B(x) then [r~z)J =[pix)] so B(z) =[Y : [¥] =[¥] =[¥]} = B(x). Let g and q be non-negative, positive homogeneous functions 1 2 on X such that q f q and q and q are not identically zero. Let 1 2 1 2 -r and -r be the eler.1entary length topologies determined from q and 1 2 1 q2 respectively. LE~1~~A 3.1. -r 1 c -r 2 if and only if for every x s X, z s B1(x) implies that B (z) c B (x) . 2 1 PROOF. -r c -r if and onlv if for every x s X, z s B (x) 1 2 1 implies z s B (v.J) c B (x) for some V.J s X. If z s B (vl) then 2 1 2 B (z) B (w) by proposition 2 = 2 3. 1. 1 1 LEMMA 3.2. If -r 1 C -r 2 then qi (0) = q~ (0) . 9 PROOF. Fori= 1, 2, Bi( O) = ly: [q s (o) (o) c s (o) 3.1, q2 1(o) c ql 1(o). and 0 B1 so B2 1 by lemma hence Suppose z s such that q (z) = 0 and q (z) f 0. Then z s B (o) so X 1 2 1 B (z) c B (o). Let 2 = m and let x E X such that q (x) 0. 2 1 [ q ~z) ] 2 ~ 2 ( Amx ) Amx q2 (y) ] [ q2 ( ) ] J Then q \q {x) = rnA so q (x) E B (z) = I y : [ A = A = m . 2 2 2 2 But B2(z) c B1(o) so q 1 (q~(~J) = 0 hence q1(x) = 0. We then have 1 1 q (x) = 0 implies q (x) = 0, since q2 (o) cql (o), and q (x) f 0 2 1 2 implies q1(x) = 0 so taking both of these results, this implies that q1 is identically zero which is a contradiction. Therefore ql1(0) = q21(0). THEOREM 3.2. T 1 c T 2 if and onlv if there is an integer k > 1 such that q2(x) = kq 1(x) for every x s X. PROOF. Let x s X such that q1(x) =A. rf [ q2~x) ] = k then q2(x) q2(x) A ~ k and, by lemma 3.2, k 1 0. Suppose that A. < k. Then AkX \ Akx ( AkX ) Ak Ak q2 ( q2{xl{ Ak so G2\x) E B2(x). But ql q2{x) = q2{x) ql (x) = q2{x) A > A implies that q~~~) t B1(x ). This is a contradiction since B (x) c B (x). Thus q (x) = A implies that q (x) = k/.. for some 2 1 1 2 integer k ~ 1. It will now be shown that the integer k does not depend on the particular x cho sen. Suppose z1, z2 s X such that q (z ) = q (z ) = A, q (z ) = A.k , q (z ) = Ak and k 1 k . 1 1 1 2 2 1 1 2 2 2 1 2 10 If > then = hence s B (z ). But k2 k1 q 2(k~:l) Ak 2 k~:l 2 2 k2zl) k2 k2zl q1( ~ = fl A> A implies that~ t B1(z 2). This is a contradic- 2 tion since B2(z2) C B1 (z2). If k1 > k2 then q 2 c~: ) = k1A implies kl z2 (kl z2) kl that - k-s B2(z 1). But q1 -c- = kA >A implies that 2 2 2 klz2 -k--- ¢ B (z ) and this is a contradiction since B (z ) c ~ (z ). 2 1 1 2 1 1 1 Hence q1(x) = A implies that q2(x) = kA for some integer k. Suppose that z s X such that q1(z) f D. Then q2(z) f 0 and q 1 ~~{zJ) = A hence q 2 (q~{zJ) = kA and therefore q2(z) = kq 1(z). km 1 11 COROLLARY 3.1. T1 = T2 if and only if q1 = q2. It is important to note here that theorem 3.2 holds if q1 and q2 are chosen to be any of the following measuring devices. 1) norm 2) pseudo-norm 3) quasi-norm 4) convex functional 5) any non-negative, positive homogeneous function. Thus if the measuring devices 1) - 5) are used, theorem 3.2 states that different measuring devices always generate different elementary length topologies. 12 IV. LOCALLY BOUNDED TOPOLOGICAL VECTOR SPACES In this chapter elementary length topologies for locally bounded topological vector spaces will be considered. The following defini tions and theorems concerning these spaces are given in [6]. DEFINITION 4.1. A topological vector space is said to be locally bounded if it has a bounded neighborhood of 0. THEOREM 4.1. The topology of a topological vector space can be given by a quasi-norm if it is locally bounded. Conversely, a quasi normed space is always locally bounded. DEFINITION 4.2. A p-norm, 0 < p ~ 1, on X is a real-valued function q on X such that Pl) q(x) >- 0; P2) q(x) = 0 => X = 0; P3) q(ax) = lalp q(x); P4) q (x+y) : q (x) + q (y) • THEOREM 4.2. A p-normed space is locally bounded. Conversely, every locally bounded space is p-normable. A construction of elementary length topologies vvith the aid of p-norms will now be given. DEFINITION 4.3. Let q be a p-norm on X. A length on X corres- pending to q is a function f : X ~ {n~ : n = 0, 1, 2, ···} such that ~11) q ( x) = n~ => f (x) = n ~; t12) => f(x ) < f(x ); q(xl) ~ q(x2) 1 - 2 1 ~ - M3) f((l x) = f(x) + ~ for x ; 0. + QTXT)Pq X 13 1 L Et~t1A 4. 1. If q(y) 2: A then f(y) = f((l- ~)Pq,y; y) +A .. PROOF. If q(y) = A the result follows from Ml) so suppose that 1 q(y) >A. Let x = (1 - y. Then q(x) = q(y) - A and ~)Pq,yJ 1 1 1 ( 1 + j) X = ( 1 + ( ) / ( 1 - y = y. By , ~)q,x; q y - 1 ::Aq,yJ l f~ 3) 1 1 f(y) = f((1 + ~)P x) = f(x) +A= f((1 - ::b)P y) +A. q \X 1 q \Y 1 For any real number r let [r] and {r} be defined as in definition G2 of chapter II. THEOREM 4.3. A function f is a length on X corresponding to a p-norm q if and only if f has one of the two forms: 1 ) f ( x ) = A[9 (x ) A- aJ where A > 0 and 0 :;: a < A; 2) f(x) = A[q(x) A- a} where A > 0 and 0 < a S A. PROOF. Suppose that f is an elementary length on X corresponding to a p-norm q and let A= {x : f(x) = 0}, a= sup{q(y) : y c A}, B(a) = {x : q(x) < a} and rr(a) = {x : q(x) s a}. It follows from t~ 1) that 0 ~ a ~ A. It is c1 ear that A c 13 (a). If x c [) (a) then there is a y c A such that q(x) < q(y) hence, by ~ · 12), x c A.. Thus B(a) c A c.B(a). Suppose that A~ B(a) .. Then there is an x c A such that q(x) = a. If q(y) = a then q(y) = q(x) and x c A so it follows from M2) that y c A. Thus A= ~ (a) or A= B(a). Suppose A= B(a). Then a~ 0 since B(O) =¢and 0 c 13(a) by t11). Thus 0 < a < A. If 1 a < q(x) 1 f (x) < f ( (q {~)) p x) = A by t·12). If A < q (x) < A + a then 1 q((l - q(~))p x) = q{x) -A < a so applying lemma 4.1 we have 1 f(x) = f((1- ::-b)P x) +A.= A. • Hence f(x) =A. for a < q(x) 1 Suppose that A+ a~ q{x) < 2A +a. Then a < q{(l - q(~)l x) = 1 q{x)- A< A+ a so f((l- q(~))p x) = A and, by lemma 4.1, 1 f(x) = f((1 - ::-b)P x) + A. = 2A. . It follows by induction that q\XJ f(x) = A(q(x)A- a} . If A = !(a) then 0 < a < A. If a < q(x) ~ A 1 1 then f(x) f 0 and q(x) < q((::-b)P x) = A. so f(x) < f((::-b)P x) = A. - q\X) - q\XJ 1 by M2). If A. < q(x) < A.+ a then q((l- ::-b)P x) = q(x) - A.< a - q \X) 1 and by lemma 4.1, q(x) = q((1 - ::-b)P x) + A. = A. . By induction, if q\X) nA. + a < q(x) ~ (n + 1)A. + a then f(x) = (n + 1) A. for n = 0, 1, 2, ••• hence f has form 1). Suppose that f is a function which has form 1) or form 2). r11) and M2) are clear and t·13) follows from the fact that 1 q((1 + ::-b)P x) = q(x) + A.. q\XJ 15 TII E0 REt1 4. 4 • Let A be a s ub set of ( 0 , oo ) vJi t h 0 as a 1 i mi t poi nt . If, for A c A, fA is a 1 ength on X corresponding to a p-norm q then lim fA (x) = q(x) for every x c X. A+O PROOF. The proof of this theorem is identical to the proof of theorem G2 of chapter II with the norm in that proof replaced uy q. DEFINITION 4.4. An elementary length on X corresponding to a p-norm q is a function f : X+ {nA : n = 0, 1, 2, ···} which satisfies t•11), M2) and 1 M3') f((n)P x) < nf(x) . THEORE~1 4 . 5. A function f is an elementary length on X corresponding to a p-norm q if and only if f(x) =A[~~x)J for every X c X. PROOF. Let f be an elementary length. It folloVJs from t.B') that 1 Let nA < q(x) <_ (n + 1)A. Then q(x) < q(((n+l) A)P x) = f(O) = 0. q(x) 1 (n + l)A so by M2) f(x) ~ f(((~(!J " lp x) = (n + l)A. Let L and m be LnA integers such that m < L and q ( X ) > m· Then, by t·12) and t1 3 • ) , 1 1 f(x) > m f(( LnA )P x) > l f(( Ln A)p x) = Ln A > nA. Thus - m m q(x) - m qrx) m f(x) ~ (n+1)A so f(x) = (n+l) A. Conversely, suppose that f(x) = A[q~x)J. r~l) and t12 ) are clear and f( (n)~ x) = ~..[q( (n/x)] = A[ng~xlJ . It is shovm in [1] that Gg~x)J ::; n [g(~)J so f is an elementary length . 16 COROLLARY 4.1. An elementary length is a length. THEOREt·1 4.6. A length f is an elementary length if and only if it is sub-additive. PROOF. Gudder•s proof of this theorem carries over without change if norms are replaced by p-norms. If f is a length or an elementary length then p(x,y) = lf(x) - f(y) I is a pseudo-metric on X. Let B(x,r) = {y : p(x,y) < r}. B(x,~) = {y : f(x) = f(y)} will be called a simple ba 11. LEMMA 4.2. Two simple balls are either disjoint or equal. PROOF. Gudder•s proof of this lemma carries over unchanged if norms are replaced by p-norms. LEMMA 4.3. Every ball is a union of a finite number of simple balls. PROOF. If x ~ 0 and B(x,n~) ~ ¢ we show that B(x,n~) = . 1 1 J=n- ·~ - .U B( ( 1 + qfx-y) P x , ~ ) . Let y E: B( x , n~ ) . Then the re i s an J=-(n-1) integer m such that 0 ~ m < n-1 and p(x,y) = m~. Suppose q(y) > q(x). 1 Then mA = p(x,y} = f(y) - f(x} andy E B((l + q{; 1)P x, A) since 1 1 p(y,(l + q{;J)p x) = lf(y) f ( ( 1 + qrxrm~ )P x) I = If (y) - f ( x) - m~ I = 0. If q(y) ~ q(x) then m~ = f(x) - f(y) and 1 1 p (y , ( 1 - q(; ) )p x ) = If (y ) - f ( (1 - q{; ) )p x ) I = If (y ) - f ( x ) + mA I = 0 17 1 . 1 1 soy E B((l- q(~ )P x, A). Hence B(x,nA) c ~Ct B((l + 1~J)Px,A). 1 J=-(n-1) q 1 '),_ - If y s B((l + ~)Pq(x) x, A) for -(n-1) < j < n-1 then 1 1 p (y, X) < I f (y) - f ( ( 1 + L)p X) I + If ( ( 1 + ~) p X) - f (X) I = - q(x) q\x 1 1 lf((l + L)Pq(x) x)- f(x)j = jjjA < n}. soy s 8(x,n}.). j=n-1 If x = 0 a similar argument shows that B(O,nA) = U B(z.,A) j=O J where f(zj) = jA. THEOREM 4.7. The balls B(x,nA) generate a non-discrete topology on X. Every (bounded) open set is a (finite) countable union of simple balls. If x;. 0 then every simple ball has the form oX+::::r:;T,o( J'Ax 1\,') J. = 0 , ±,1 ±, 2 •••. q\XJ It is natural to ask if different p-norms can generate the same elementary length topologv. Some preliminary results are needed to answer this question. We first note that if B(x) = B(x,A) is a simple ball defined with the aid of an elementary length f defined in terms of a p-norm q then B(x) = {y : f(x) - f(y)} = (Y : [g~x)J = [9.fl]) . PROPOSITION 4.1 . If x s X and z s B(x) then B(z) = B(x). PROOF. If z E B(x) then [q~x)J = [g~z)J . Hence B(z) = [Y : [~] = [~] = [~]} = B(x). 18 Let q and q be p-no rm s on X VJ 11 i c h are not i dent i c a 1 1y zero an d 1 2 are such that, for every scalar a and every x s X, LEt 1t~A 4 • 4 • If s 1 = {X : q 1 (X ) < A. } and s') = {x L then 51 n 52 f { 0}. PROOF. The function q is not identically zero on X so there is 1 1 ~ A. P1 a z s X such that q (z) f 0. It fo 11 0\'/S 1 that q1((q (z)) z) = .\ 1 hence s "f {O}. Let y s s such that y f 0. If q (y) ::; .\ vJe are 1 1 2 finished so suppose q 2 (y) > .\ . Let B be a real number such that 1 .\ p2 0 < B < (q (y)) < 1. Then q (Sy) 2 1 Let T and T be the elementary length topologies determined VJith 1 2 the aid of q and q , respectivelv. As in the proof of le111ma 3.1 of 1 2 chapter III, the proof of the next lemma follows from proposition 4.1. LE ~ 1MA 4 • 5 . T c. T i f and on 1y i f f o r eve r y x s X, z s B ( x ) 1 2 1 implies that B (z) C B (x). 2 1 THEOREM 4.8. If q1 f q2, then Tl f 1 .\ pl p R0 0 F . CAS E 1 . Sup po s e ( q (y ) ) f 1 19 1 1 A P1 A P2 P1 A ( q (y ) ) < a < ( q ( y ) ) . Then q1 ( ay ) = a q 1 ( y ) > q ( y ) q 1 (y ) = A 1 2 1 · P2 A and q (ay) = a q (y) < q (y) q (y) = A. By lemma 4.4 there is a 2 2 2 2 = w •• [ q 2 ; ) ] = z s s1 n s2 such that z f o. Then a y s B2(z) I A ~I 2 [ q ;z)] = 1) and a y 13 (z) since q (ay) >A. Thus t 1 1 z s B (z) B (z) CB (z) so -r (/;-r by lemma 4 . 5. If 1 r> 2 1 1 2 1 1 A P2 A P1 (q (y)) < (q (y)) a similar argument shovJs that -r 2 tt:-r 1• 2 1 1 1 - ~ A P1 A P2 CASE 2. Suppose that x r 0 implies that (q (x)) = (q (x)) 1 2 If p p then q = q so assume that p < p . It is easy to verify 1 = 2 1 2 2 1 It is 1 1 p2 q1(x) p-1 q2(x) p-2 ql(x) p-1 q2(x) also true that ( A ) = ( A ) if and only if ( A ) = A q 1 (x) It follows that if m > 1 then (m - 1) < A ~ m if and only if Since q is not identically zero there is a 1 1 2A -P1 y s X such that q (y) r 0. If z = (q (y)) y then q (z) = 2A. 1 1 1 20 p2 q1(z) q2(z) P1 Hence 1 < A < 2 and this implies that 1 < A ~ 2 < 2. If 1 2A p2 q2(az) -- [q2A(z)] and a = then---,----= 2 this implies that q2(z) 1\ p2 q2 (az) p1 But --!:A----= 2 > 2 and this implies that a z i B1 (z) p2 q1 (x) q2(x) P1 since 1 < A ~ 2 if and only if 1 < A ~ 2 Hence B2(z) d: B1 (z) and it follows from lemma 4.5 that T 1 t:C T 2• If p1 < p2 a similar argument shows that T2 ~ T1. The topology of a locally bounded topological linear space may be given by a p-norm or a quasi-norm. It is therefore natural to ask if a p-norm and a quasi-norm can generate the same elementary length topology. The answer is contained in the next theorem which deals with a more general case. THEOREM 4.9. Let q1 be a p-norm, q a non-negative, absolutely homogeneous function and let T1 and T be the elementary length I topologies on X determined by q1 and q respectively. If q q1, then -r 1 ~ T. PROOF. We consider two cases: 1 1 1 CASE 1. Suppose {0} = qi (o) ~ q- (0) and let y s q- (0) such that 1 1 y "f 0. If z = (q (y )y y then q ( z) = A• Now, q( z) = 0 so 0 E B( z) 1 1 but 0 t B1 (z) hence B(z) ~ B1 (z) and it follows t hat T 1 1 1 CASE 2. Suppose ql (o) = q- (0). Since the proof of theorem 4.8 did not require sub-additivity the proof of theorem 4.8 with p2 = 1 may be used for this case. 22 V. LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES A. LOCALLY CONVEX SPACES AND TIKHONOV SEMI-FIELDS The following definition and theorems are given in [5]. DEFINITION 5.1. A topological vector space is locally convex if it has a base at 0 consisting of convex sets. THEOREM 5.1. The topology of a locally convex space X can always be defined by a family of pseudo-norms on X. THEOREM 5.2. If X is a locally convex topological linear space with topology generated by a family P = (p ) as 6 of pseudo-norms, a then X is a T1 space if and only if to each x ~ 0 corresponds pas P such that pa (x) ~ 0. We now state some definitions and results which are given in [7]. Let 6 be an arbitrary set and let R6 be the set of all real functions f 6 ~ R. For q s 6 and a,b s R let Uqa, b be the set of all 6 f s R that satisfy the condition a < f(q) < b. Taking the sets Uqa, b as a subbase we obtain a topology for R6 and R6 with this topology is called a Tikhonov semi-field. In what follows, for any set 6, R6 will always denote a Tikhonov semi-field. 6 6 Let K be the set of all f s R such that f(q) > 0 for every q s 6. The set K6 is called the cone of strictly positive elements 6 6 and its closure K consists of all functions f s R for which f(q) > 0 for all q s 6. Addition and multiplication in R6 are defined 6 coordinatewise: for f,g s R and q s 6, (f+g)(q) = f(q) + g(q), 6 (fg)(q) = f(q)g(q). Thus an order may be defined on R by f > g iff 6 f - g s K . 23 DEFINITION 5.2. Let X be an arbitrary set. A mapping p : X x X+ K~ is a metric on X over R~ if the following conditions are satisfied: 1) p(x,y) = 0 if and only if x = y; 2) p(x,y) = p(y,x); 3) p(x,y) p(x,z) <- + p (y' z) . R~ 6 A set X with a metric p over 1. s called a metric space over R . THEOREM 5.3. Let (X,p) be a metric space over R~, U a neighbor hood of the origin in R~, and x s X. If ~(x,U) = {y s X : p(x,y) s U} then the family of sets ~(x,U), as U varies over the neighborhoods of the origin in R~, can be taken as neighborhoods of x which introduces a topology on X. This is called the natural topology of X. For f s R6 define If! by If! (q) = !f(q)!. THEOREM 5.4. Let ~ be an arbitrary set. The mapping p : R~ x R~ + K~ defined by p(f,g) = !f-g! makes R~ into a metric space over R~. The natural topology of this metric space coincides with the initial topology on R~. The following definition is given in [8]. DEFINITION 5.3. Let X be a linear space over the real or complex numbers. A mapping II II X ~ K6 is a norm on X over R~ if it satisfies the following conditions: 1 ) II x II = 0 i f and on 1y i f x = 0 ; 2) II ax II = Ia I II x II for every sea 1a r a and every x s X; 3) II x+y II 2 II x II + II Y II · If II II is a norm on X over R~ it is easy to see that p(x,y) = I lx-yl I is a metric on X over R~. If the topology of a 24 topological vector space X is the natural topology of X as a metric space over R~ and the metric is defined in terms of a norm over R~ then X is said to be normable over R~. The function II II of definition 5.3 will be called a pseudo norm over R~ if it satisfies 2) and 3) but may be zero at some x f 0. It is clear that the definitions of pseudo-metric over R~ and pseudo-normable over R~ may be defined by making the appropriate alterations of the definitions of metric over R~ and normable over R~. By theorem 5.1, the topology of a locally convex topological vector space X can be defined by a family (p a )aso A = P of pseudo-norms defined on X. As in [8], we define II II : X -+ K~, where ~ is the as~ s index set of the family P, by I lxl l(a) = pa (x) for and x X. It is easy to verify that this mapping is a pseudo-norm on X over R~. It follows from theorem 5.2 that this mapping is a norm on X over R~ if X is a T1 space. These results are summarized in the following theorem due to Iseki [8]. THEOREM 5.5. Any locally convex topological linear space X is pseudo-normable over R~ for some set~- If X is a T1 space then it is normable over R~ for some ~- B. ELEMENTARY LENGTH TOPOLOGIES FOR LOCALLY CONVEX SPACES Let (X, II II) be a locally convex space where II II is a pseudo-norm or a norm over R~. DEFINITION 5.4. A length on X over R~ is a function f X -+ R~ such that for any a s ~ 25 N1) llxll (a) = nA => f(x)(a) = nA; N2) llx111(a) 2 llx211(a) => f(x1)(a) :: f(x2)(a); N3) f(x + llxll(a) x)(a) = f(x)(a) +A if llxll (a)> 0. DEFINITION 5.5. For any real number r define [r] and {r} as in definition G2. Then let [ ] : R~ ~ R~ and { } : R~ ~ R~ be defined by [f](a) = [f(a)] and {f}(a) = {f(a)}. THEOREM 5.6. A function f : X~ R~ is a length on X over R~ if and only if f has one of the two forms 1) f(x) = f [ llxll- 9 ] where f (a)= A > 0 and 0 < g(a) 2 ) f ( x ) = fA { II x I} A- 91where fA( a ) = A > 0 and 0 < g (a ) < A for every a s ~. PROOF. Suppose that f is a length over R~ and as ~. Since llxll (a) = p (x), for each as ~ the conditions N1), N2) and N3) of a definition 5.4 imply that f(x)(a) is a length on X defined in terms of Pa· It then follows from theorem 3.1 that f(x)(a ) =A[pa(x) A- aa J where A > 0 is fixed and 0 S aa < A or f(x)(a ) =A {pa(x) A- aa Iwhere 0 < aa 2 A. Now define fA s R~ by fA( a ) =A for every as ~ and define g s R~ by g(a ) =a . Then f has form 1) or form 2). a If f has form 1 or form 2 then, by theorem 3.1, for each as~ f(x)(a) is a length on X defined in terms of llxll(a) = p (x) . It a follows that N1), N2), and N3) of definition 5.4 hold. The statement of the next theorem requires the following concepts which can be found in [7]. 26 Let (X,p) be a metric space over R~ and H a directed set. A mapping x : H +X is called a sequence of type H on X. The image x(~) of ~ E H under x is denoted by x~ and the sequence is written in the usual form x = (x~), ~ E H. A sequence x of type H in X over R~ is said to converge to y E X if the following condition is satisfied: for every neighborhood of the origin U c R~ there is an element ~U E H such that p(x~,y) E U Convergence is denoted by writing x~ + y or lim x~ = y. ~EH In other words, lim x~ =yin X if and only if lim p(x~,y) = 0 in R~. ~EH ~EH Let A = {fA E R~ : A E (0,1) and fA (a) = A for every a E ~}. The set A is directed by the relation fA << fA if and only if 1 2 A2 S A1. Also, for each A E (0,1) and each fixed x EX, the mapping fA+ f(x), where f(x) has form 1) or form 2) of theorem 5.6, is a sequence of type A on R~. THEOREM 5.7. Iff is a length on X over R~ for each fA E A then lim f(x) = llxll for each x EX. fAEA n a. 1 ~ PROOF. The set U = n U = {h E R : E > 0, a; E ~ and i =1 E -E < h(a.) < E for 1 < i < n} is a basic open set about 0 in R~. It 1 - suffices to show that for each such U there is a (fA)U E A such that p(f(x), llxlll = lf(x)- llxlll £ U for all fA£ A such that fA>> (fA)U. For each a;, 1 s i < n, lf(x)- JJxJJI(a;) = lf(x)(a;)- JJxJJ(a;)l = lf(x)(a;)- Pa;(x)l 27 where f(x)(c.i) = A[peti (x) A- aCt; lor f(x) (a;) = Arai {x) A- aai I· Then, by theorem 3.1, f(x)(a.) ~ p (x) as A~ 0 for 1 < i A < 8 .• 1 Choose AO € (0,1) such that AO < min {8.} then let 1:si:sn 1 lim f(x) = llxll fA€A DEFINITION 5.6. An elementary length over R6 is a function f : X~ R6 which satisfies N1), N2) and N3') f(nx) 2 nf(x) for every x € X and n = 0, 1, 2, •••. THEOREM 5.8. A function f : X ~ R6 is an elementary length over Rll if and only if f(x) = fA [ I}~II ] . 6 PROOF. Let f be an elementary length over R and a € 6. The mapping f X~ R6 defined by f (x) = f(x)(a) satisfies a a N1) llxll (a) = pa(x) = nA => fa(x) = nA; N2) llx111 (a) = pa(x1) S llx211 (a) = pa(x2) => fa(x1) < fa(x2); N3 • ) f ( nx) < nf ( x). a - a It follows from theorem 3.1 that f (x) = f(x)(a) is an elementary a length on X defined in terms of the pseudo-norm pa(x) and that f (x) = f(x)(a) = A[llxll (a)]= f [llxll] (a). Thus f(x) = f [ llxll ]· a A A fA A fA 28 If f(x) =fA[! I~~~ J then f(x)(a) is an elementary length on X defined in terms of p (x). By theorem 3.1 f(x)(a) satisfies N1), a N2), and N3') above so f is an elementary length over R.b. COROLLARY 5.1. An elementary length over R~ is a length over Rb.. DEFINITION 5.7. A function f : X~ Rb. is sub-additive if f(x+y) :: f(x) + f(y). THEOREM 5.9. A length f over Rb. is an elementary length over Rb. if and only if it is sub-additive. PROOF. If f is a length over Rb. which satisfies f(x+y) < f(x) + f(y) then, by induction, f(nx) < nf(x) for every X E: X. If f is an elementary length over Rb. then f(x) = f [ ' lxl A fA I] so f(x+y) = f [ llx+yll J < f [llxll + It is shown in [4] A fA - A fA IIYII ]. that for real numbers r,s : 0 [r+s] ~ [r] + [s]. Then f(x+y)(a) < fA [IIxll ;AllYl! ] (a)= A[l!xll(a) ~ IIYII(a) J ~ A( II x{I (a l J + A[ II YI{ (a l J= ( fA [11 ~~I ] + fA [llf ~I ]}a) so f(x+y) :: f(x) + f(y). COROLLARY 5.2. If X is a locally convex topological linear space and f is an elementary length on X over Rb. then p(x,y) = f(x-y) is a pseudo-metric on X over Rb.. If X is a T1 space then p(x,y) = f(x-y) is a metric on X over Rb.. 29 PROOF. If x = y then p(x,y) = fA[~A] = 0 and p(x,y) = p(y,x) follows from 11-xll = llxll. p(x,y) = f(x-y) = f((x-z) + (z-y)) < p(x,z) + p(z,y). If f(x-y) = fA [ IIxf~ll ] = 0 then llx-yll = 0. If X is a T space it follows from theorem 5.2 that l lx-yl l 0 1 = if and only if x = y. The natural topology for the space (X,f(x-y)) will now be considered. Suppose that there is an aj E 6 such that Pa. E P is a J a. norm. Let x EX and consider ~(x,U) where U = uAJ. rl(x,U) =l y: f(x-y) = fA l llxf~ll ] E u:j } = ly :A [ IIx-y~l(aj) ] Iy : [ Pa Hence ~(x,U) = {x} so the natural topology is discrete in this case. A further discussion of these topologies requires the following definition which is given in [5]. DEFINITION 5.8. A family (p )a E 6 of pseudo-norms is saturated a if for any finite sub-family {p , ···, p } the pseudo-norm p al an defined by p(x) = max IP (x) } also belongs to the family. l~j.~n aj TH EOREM 5.10. If the family (p )a E 6 is saturated and for a each a E 6 p fails to be a norm then the natural topology for a (X,f(x-y)) is not discrete. PROOF. We show that every neighborhood ~(O,U) of 0 contains 30 n a. some y :f 0. It is sufficient to take U = n U 1 since sets of the i=1 A form ~(x,U) are a base at x for (X,f(x-y)). Now, r:l(O,U) = ly : A[ pa (p )a s 6, p fails to be a norm. This means that there is some y s X a such that p(y) = 0 and y :f 0. Thus pa; (y) = 0 for 1 <-- i < n so y s ~(O,U). THEOREM 5.11. (X,f(x-y)) is a Hausdorff space if and only if (X, II II) is a Hausdorff space. PROOF. Suppose that (X,f(x-y)) is a Hausdorff space and x s X such that x :f 0. If pa (x) = 0 for every a s ~' then I lxl I = 0. Hence f(x-0) =p(x,O) =fA [ ! I~~~ ] = 0 and therefore p(x,O) £ U for every neighborhood U of the origin in R~. This is a contradiction since, in this case, {O,x} s ~(x,U 1 ) 0 ~(O,U 2 ) for every choice of u1 and u2. By theorem 5.2, (X, II II) is Hausdorff. Suppose that (X, II II) is Hausdorff and x ,y s X such that x-y :f 0. By theorem 5.2, there is an a. s ~such that llx-yll (a .) = a· J J Pa.(x-y) :f 0. Let U = uAJ and let Q(x,U), ~(y,U) be neighborhoods J of x andy in (X,f(x-y)). Suppose that z s ~(x,U) (\ ~(y,U). Then llx-zll (a.)] (11z-yll (a.)] f(x-z)(aj) =A ( A J = 0 and f(z-y) = f(y-z) =A A J = 0 so II x-z II (aj) = II Z-YII (aj) = 0. But this means that 31 Pa.(x-y) = 11x-y11 (o:j) ~ 11x-z11 (aj) + llz-yll (o:j) = 0 v~hich is a J contradiction. Thus r2(x,U) and r2(y,U) are disjoint neighborhoods of x andy so (X,f(x-y)) is Hausdorff. EXAMPLE 5.1. Let r2 ~ ¢ be an open subset of Rn and let C(~) be the linear space of all continuous functions from Q into the complex numbers. For every compact subset K of r2 the function q K defined by qK(f) =max lf(x)l is a pseudo-norm on C([2). The family (qK)' \vhere XEK K runs through the compact subsets of~, generates a locally convex topology on C(n). It is easy to shmv that this family is saturated and we now show that each qK fails to be a norm. If K is a compact subset of Q then Q'K is an open subset of n. Let d(x y) = lx-yj be the usual metric for Rn, x E Q,K and \ (x ) {x: lx-x 1 < r } 0 0 = 0 0 0 where ro is chosen so that sro(xo) c Q'K. Define f E c(n) by if I x-x 0 I < r 0 f (x) = 0 Since f(x) = 0 on K, qK(f) = 0 hence qK fails to be a norm. If f E C(n) such that f(x ) t- 0 then q{x }(f) ~ 0 so C(n) is a Hausdorff 0 0 space. Then by theorems 5.10 and 5.11 the space (C(n),f(x-y)) is non-discrete and Hausdorff . EXAMPLE 5.2. The space of distributions on an open subset of Rn, as described in [9], also satisfies the conditions of theorems 32 5.10 and 5.11. The verification is rather lengthy and will be omitted. An elementary length topology which is Hausdorff does not correspond very well with the properties that might be expected of such a topology. In the normed linear space case the elementary 1ength topologies on X are always non-T0• A development along the same lines as the one for the normed linear space case will now be given. In \vhat follows, a length over R~ or an elementary length over R~ may be used. For convenience, an elementary length v.Jill be chosen. Let f be an elementary length over R~. The mapping d: X x X~ K~ defined by d(x,y) = if(x)- f(y)i is a pseudo-metric on X over R~ and the natural topology for X determined by d will now be considered. A base at x s X consists of sets of the form ~ (x,U) n a. 1 where U = n U • Throughout the remainder of this chapter, 1=. 1 s elementary length topology Hill mean the natural topology of the pseudo-metric d on X over R~. THEOREM 5.12. The space (X,d) is never T0. PROOF. Let x s X such that x 1 0 and let y = ax \1/here a 1 1 is a scalar such that lal = 1. Now, d(x,y) = jf(x)-f(y) I and f(y) =fA[~]= fA[ll~~~] = f(x) so d(x,y) = 0. But then {x} and {y} have the same closure in (X,d) so this space is not T0. If (X,-r) is a locally convex space such that -r is generated by a family (pa)a s ~ of pseudo-norms, and p6 is a continuous pseudo-norm 33 on X then (p ) AU {p~}, where o t 6, also generates the topology T. a aE:u u In other words, the topology of X may be generated by pseudo-norms over different semi-fields. It seems natural to call two such pseudo-norms equivalent and to ask if the elementary length topologies generated by equivalent pseudo-norms over semi-fields are comparable. A necessary condition for the simple case described above will now be given. Suppose (p a ) aE:uA is a family of pseudo-norms which generates the topology T and p0 is a continuous pseudo-norm on (X,T). Let t 1 and t be the elementary length topologies generated by (p ) A and 2 a aE:u (pa)aE:£1 U {p0} respectively. Let S = {x : p0(x) = A}. THEOREt~ 5.13. If t C t then for each x E: X such that p (x) 0 2 1 0 t- there exist a finite set JxC/1 and functions j : S-+ Jx, I< : S-+ AX {1, 2, 3, ···} such that Pj(vJ)(x) = K(vJ)p 0(x) where w = PIXT E: S. 06 62 be R PROOF. Let II 11 1 and II 11 2 pseudo-norms over R and respectively where 6 = 11 U {o}. The elementary length topologies 2 t 1 and t 2 are generated by the pseudo-metrices = f, [ II xf; l2 and d2(x,y) 1\ 1\ J _ If x E: S then, since t c t , there is a finite subset 2 1 {y : d (x,y) E: U } where For each j E: Jx, 2 2 u1 let mj. [Pa1(x)J = If m. = 0 for every j E: Jx then Pa.(2x) = 0 so J J 34 Pa.(x)l [Pa.(y) l ) y : I 2x s n (x,U = ~ = ~ for every j s Jx • But 1 1 I [ p (2x) = 2A so 2x ¢ S< (x,u ) = Po;Y)] = [Po;x) ] = which 6 2 2 ly : [ 11 j s Jx. Suppose that p (x) < m.A for every j s J such that Ct. J X J P (x) a. A m. Pa. (x) ~ 0. If Pa. (x) ~ 0 then mj-1 < ---!i-- < mj so 1 < p 1x) . J J Ct. J It then follows that 1 < b = minl:a:1x) : j E Jx and Paj (x) f 01. Let 1 < S < b. If p (x) = 0 then p (Bx) = 0 and if p ( x) ~ 0 a. a. Ct. J J J pa . ( x ) B pa . ( x ) pa . ( Bx ) b pa . ( x ) for j s Jx then mj - 1 < ~~~- < __J~A- = __;,:.J~A~ < -...... :--i- < A m. p (x) J a. Pa.(J A = mj so S x E n1(x,U 1). But p0 (Sx) =SA > A so J B x t ~~ (x,U ). This is a contradiction since n (x,u )C ~ (x,u ). 2 2 1 1 2 2 Thus if p~(x) = A then p (x) = m.A for some j s J . Hence for x s S u Ctj J X there is a mapping j : S ~ Jx c ~and a mapping K : S ~ {1,2,3,···} X such that pj (x) (x) = mj (x)A = K(x)p0(x). If z s such that p0 (z) f 0 then w = p:fzl E S hence pj(w)(p:tzJ) = K(w)A. Therefore Pj(w)(z) K(w)p (z) where j and K are the functions described above. = 0 35 THEOREM 5.14. Each of the following conditions is sufficient for t2 c t1 . 1) There is an integer k > 0 and an a s 6 such that pa (x) = kp (x) 0 for every x s X. 2) There is a finite subset J of 6 such that p0(x) = max {pa(x)} asJ for every x s X. PROOF . Suppose 1) holds and let d (x,y) and d (x,y) oe defined 1 2 as in the proof of theorem 5. 13 . If U and V are neighborhoods of the 6 origin in R then r2 2(x,U)f1 r2 2(x,V) = {y: d2(x,y) s U} fl {y : d ( x , y ) s = r2 ( x , u Thus i f u = u i t s uf f i c e s to s ho v1 2 v} 2 nv ). 2 ~ that r2 2(x,U 2) s t 1. For if r2 2(x,V 2) is a t 2 basic open set about x, then we may take v = A U ~ where I is a finite subset of 6. If 2 Ssi o t I then r2 2(x,V 2) s t 1 and if o s I then r2 2(x,V 2) = r2 2(x,U 2) n r2 2(x,V2) where V2 =(\{U ~ : S s I, S t- o}. Hence if r2 2(x,U 2) s t 1 vJe have r2 2(x,V 2) s t 1 since r2 2(x,V2) s t 1. Let Ta and T0 be the elementary length topologies determined vdth the aid of pa and p0 respectively. ~y theorem 3. 2 To c Ta. If u1 = u ~ then 0 Suppose that 2) holds and let u2 = u ~._. l-J e s hOVJ that n2( x,U2) s t1 for every x s X. Let U1 = n { U ~ : a s J} an d 36 : a s J} there is a S s J such that as J. Since p0 (x) = max{pa(x) p(x) = p (x) so m= [p < y E [Po ~x) ] = ms l· Now, m m0 for every a s J and for every J such a - f.J px(y) that my = m 'v'Je have m - 1 < A ~ E J rna f 8 13 ms. If a such that ms Pa (y) Ps (y) < m < m - 1 < < m . It follows that then rna ~ m8 - 1 so A a - S A - S Po (y) m - 1 < A < m soy s ~ (x,U ). Then ~ (x,U ) C ~ (x,U ) so 8 8 2 2 1 1 2 2 ~ 2 (x,U 2 ) E t 1• Let ( pa ) asuA be a fami 1y of pseudo-norms on X and 1 et II II be an ordinary norm on X. Let -r be the locally convex topology generated by the family (p ) A" If, for each as!:., there is an integer k a aEu a such that k p (x) = llx II for every x E X, then (X,-r) is normable. a a Thus the next theorem shows that the elementary length topology for a non-normable locally convex space cannot be determined from Gudder•s construction for normed linear spaces. Let t 1 and t 2 be elementary length topologies on X generated with the aid of (p a ) asuA and the ordinary norm II II respectively. THEOREM 5.15. If there exists a E t:. such that for every positive integer k the ordinary norm, II II, is not equal to k pa then t 1 f t 2. PROOF. We prove the contrapositive of the theorem: If t 1 = t 2 then for every a s t:. there is a positive integer k such that a ka pa(x) = llxll for every x s X. If t 1 C t 2 then for each a s t:. 37 let Ua = U~. Then for every x s X n(x,Ua ) s t 1 = t 2. Hence if ta is the elementary length topology generated by applying the Gudder construction in terms of p we have t c t since a a 2 n(x,U ) B (x,A) __ ly ·. [ PaA(y)] __ [PaA(x)]) . Then by theorem 3.2 = a I lxl I = ka pa (x) for some positive integer ka and every x s X. 38 VI. TOPOLOGICAL LINEAR SPACES Let (X,t) be a topological linear space. We recall a basic theorem [5]. THEOREM 6.1. There exists a fundamental system U of neighborhoods of 0 such that: I. Each U in U is balanced and absorbing; II. If U s U and at 0, a U s U; III. If U s U, there exists V s u such that V + V CU. The following two theorems, due toT. L. Hicks [10], are reformu lations of results given by D. H. Hyers in [11]. They make possible a unified approach to the study of topological vector spaces; that is, we can view the topology as being given by a "measuring device" over a Tikhonov semifield. THEOREM 6.2. Let u be as in theorem 6.1. If Us U, Pu denotes the Minkowski functional of U. 1. Pu(O) = o and Pu(x) > o. 2. Pu (ax) = Ia I Pu ( x) . 3. Given n > 0 and U s u, there exists o > 0 and V s U such that Py(x) < o and Py(Y) < o implies Pu(x+y) < n. 4. U ::> V (Us V) implies Pu(x) s Py(x). 5. If (X,t) is Hausdorff, Pu(x) = 0 for every U s U implies X = 0. THEOREM 6.3. Suppose ~ is a directed set and {pu U s ~} satisfies conditions 1- 4 of theorem 6.2. For each Us~ and r > 0, 1 et 39 Then there exists a unique topology t for X such that X is a topological linear space and {Su(r) : r > 0, U s 6} is a funda mental system of neighborhoods of 0. Let 6 be a directed set and (X,t) a T2 topological linear space . 6 DEFINITION 6. 1. A function II II X+ K is a semi-norm over R6 if it satisfies 1) llxll = 0 if and only if X = 0; 2) II sx II = IB I llxll for every scalar S and every X E X; 3) if a1,a2 s 6 such that a1 > a2 then llxll(a1) > llxll(a2); 4) given s > 0 and a1 s 6 there exists o > 0 and a2 s 6 such that if llxll (a2) < o and IIYII (a2) < o then llx+yjj (a1) < s . Let U be as in theorem 6.1 . For U,V s U define U ~ V if and only if U ~ V. It is easy to verify that (U, ~ ) is a directed set. If 6 = U the mapping II II: X+ RL defined by llxii(U) = Pu(x), where Pu is as in theorem 6.2, is a semi-norm on X over R6 . The next result now follows from theorem 6.3. THEOREM 6.4. The topology of any T2 topological linear space 6 may be generated by a semi-norm over R for some set 6 . Theorems 5.6, 5.7 and 5.8 remain unchanged if the pseudo-norm 6 over R6 in that development is replaced by a semi-norm over R . If f(x) = fA[~] is an elementary length over R~ corresponding to a 6 semi-norm over R then d(x,y) = jf(x)- f(y)l is a pseudo-metric on 40 X over R6. The natural topology on X determined from d(x,y) will be called an elementary length topology on X corresponding to the semi norm, II II , over Rll. The topology of a T2 topological linear space X may be generated by a sem1-norm. over Rll . The set ll is a fundamental system of neighborhoods of 0 in X. Thus a fundamental system ll 1 might be chosen lll and the same topology on X generated by a semi-norm over R . If we suppose that ll = ll U {o}, where o ¢ ll , and that ll is equivalent to 1 1 ll, it is natural to ask if the elementary length topologies T and Tl generated with the aid of the semi -norms II II and II 11 over Rll 1 61 and R respectively are comparable. If S = {y : I IYl 1 (o) =A} we 1 have the following necessary condition for Tl CT. THEOREM 6. 5. If T T then for each x s X such that 1 c I lxl ! 1(o) f 0 there exists a finite subset Jx of ll and functions j : S -+ Jx, K : S -+ {1, 2, 3, ···}such that II x II (j ( w) ) = K( w) II x 11 1 ( 0 ) where w = II x 11 ~ (0 ) E S• PROOF. The proof of theorem 5.13 may be carried over without significant change. THEOREM 6.6. Each of the following conditions is sufficient for Tl C T. 1) There is an integer K > 0 and an as ll such that I lxl l(a) = 1 (o) Kl lxl 1 for every x s X. 2) There is a finite subset J of ll such that II x 11 ( o) = max { II x II (a )}. 1 asJ 41 PROOF. The proofs of 1) and 2) of theorem 5.14 may be modified in an obvious way to prove this theorem. REMARK. The requirement that the topology of a topological linear space be Hausdorff may be dropped and the appropriate modifica tions of the results presented in this chapter remain valid. In the absence of the Hausdorff requirement, the semi-norm over R~ described earlier may be zero for some x r 0. It seems appropriate to call such a function a pseudo-semi-norm over R~. Then any topological linear space is pseudo-semi-normable over R~ for some set~ and an elementary length topology may be defined with the aid of this function. 42 VII. QUANTUM MECHANICS USING ELEMENTARY LENGTH TOPOLOGIES In [3] Atkinson and Halpern gave the following quantum mechanical description of the scattering of two particles of energy and momentum . E1,P1 and E2,P2 Let K be the complex numbers, Xr the reals with an elementary length topology and C(Xr,K) the set of all bounded, continuous functions from Xr into K. Define D : C(X r ,K) + C(X r ,K) by (Df)(x) = f(x+A) 2 ~ f(x-A) . Corresponding to the momentum is the operator P = -i~D which is self-adjoint with respect to the inner 00 product on C(Xr,K) defined by (f,g) =A L f * (nA)g(nA) where f * is n=-oo the complex conjugate of f. A position operator is defined by 1 1 1 1 1 i - 1 ( A )) q(x) =A~.[ J Then ~(p ,q ) = g(p ) exp (~sin ~ is a 1 1 1 solution of the equation -i~OW = p ~. The function ~(p ,q ) is then used to draw certain conclusions concerning the scattering of the particles. In this chapter a development similar to the one described above will be given for certain linear spaces. Let X be a topological linear space over the real or complex numbers such that the topology of X is determined by a single real valued function q; i.e., a semi-norm, norm, p-norm, quasi-norm or a convex functional. If B(x) = (Y : [.9}il-J = [q~xlJ} then {B(x) : x € X} is a base for an elementary length topology on X. Let X denote X with this topology. The following results will be used r repeatedly in the remainder of this work. 43 PROPOSITION 7.1. f s C(Xr,K) if and only if for each x s X, f is constant on B(x). PROOF. Suppose that f s C(X ,K). For r > 0 let r Sr(f(x)) == {z: lz-f(x)j < r}. Then since f is continuous there is an open set G about x such that f(G) c Sr(f(x)). Since G is open x s B( y ) C G f o r so me y c X• But by pro po s i t i on 3 • 1 13 ( y ) = B( x ) so f(B(x)) C Sr(f(x)) for every r > 0. Suppose that there is a y s B(x) such that f(y) f f(x). Let r = jf(y) 2 f(x)l Then 0 [f(y) - f(x) I = 2r > r so f(B(x)) sr (f(x)) 0 0 ct; since 0 f (y ) t S ( f ( x ) ) . Thus f i s constant on B( x ) . ro Suppose that f is constant on B(x) for every x c X and that y s X. Let G be an open set containing f(y). If y s B(x ) then n f(B(x )) {f(y)} CG so f is continuous at y. n c For each positive integer n let x n s X such that q(x n ) = n ~ and let x -n = -xn. For f s C(X r ,K) the terms of the expression 00 2 ~ jf(x )1 do not depend on the choice of x since, by proposition n=-ooL n n 7.1, f(xn) == f(xn') for any x' such that q(x') = q(x) = n. 1\lso, n n n if y s B(x) then -y s B(x) hence, by proposition 7.1, f(y) = f(-y) and, in particular, f(x ) f(-x ) = f(x ). Thus -n n n PROPOSITIOtj 7.2. E = \f f c C( X , K) and I I f ( x ) ff\01 r n=- oo n is a subspace of C(Xr,K). 44 PROOF. Let f,g s E. Then 2 + 2 I lf(x )1 + 4 I lf(x )g(x )I n=l n n=l n n 00 + 4 L lf(x )g(x )I n=l n n 00 so it suffices to show that L lf(x )g(x )I < oo. By Holder's n=l n n 00 00 2 ~ 00 2 ~ inequality for series L lf(x )g(x ) I < L lf(x ) I 1( L lg(x ) I )1 n=l n n - ( n=l n ) n=l n and this product is finite since f,g s E. For f s E and as K, a f s E is clear. PROPOSITION 7.3. ( , ) : Ex E-+ K defined by 00 * (f,g) = A \ f (x )g(x ) is an inner product on E. L n n n=-oo 2 2 lf*(x )g(x )I < lf*(x )1 )~( lg(x )1 )~ by Holder's n=lI n n - (n=lI n n=lI n 45 inequality for series. This product is finite since f * ,g E E therefore 00 the series I f * (x )g(x ) converges. Thus (f,g) E K. If (f,f) = 0 n=1 n n Since f is constant on B(x ) it then f(xn) = 0 for every integer n. n follows that f(x) = 0 for every x E X. The remaining requirements are straightforward. DEFINITION 7.1. Two normed linear spaces Nand N1 are isometri- cally isomorphic if there exists a linear transformation T : N ~ N1 such that Tis one-to-one, onto and I IT(x)l I = I lxl I for every x EN. PROPOSITION 7.4. E is isometrically isomorphic to £2. PROOF. Let {x }00 be a sequence of vectors in X such that n n=O q(xn) = n\. Define T : E ~ £2 by T(f) = (/\f(x0),/Zrf(x1),/Zrf(x2),···). 2 2 2 Since ~ lf(x 0 )1 + 0I 1 2Aif(x 0 )1 )~ =(A0 I_oolf(x 0 )1 )~ = I lfl I, 00 1 Suppose {x } is a sequence such that q(x~) = n\. For n n=O each n, x•n E B(x n ) so it follows from proposition 7.1 that f(x~) = f(xn). Thus T(f) is independent of the sequence {xn} and therefore T is well defined. 2 ••• Given x E X, x E B(x ) for ' . n exactly one value of n. Since f 1, f2 E C(Xr,K), it now follows from proposition 7.1 that f 1(x) = f 2(x) for every x EX. Therefore Tis one-to-one. If (z0, z1, ···) E £2 define f : Xr ~ K by 46 lz if X € B(O) If 0 f(x) = 1 -z if X € B(x ), where q(x ) = n:A, n = 1, m n n n 2' By proposition 7.1, f s E and it is clear that T(f) = (z0, z1, ···). Therefore T is onto. T(f +f ) = (/\ f (x ) + lr f (x ), ;zr f (x ) + /Zr f (x ), ···) = 1 2 1 0 2 0 1 1 2 2 T(f1) + T(f2) and T(af) = aT(f) soT is linear. 2 2 2 IIT(f)ll = (Aif(x 0 )1 + 2A nL lf(xn)1 t =(: j_""lf(xn)1 )!;; = 1\f\\. Therefore T is an isometric isomorphism. PROPOSITION 7.5. (E,( , )) is a Hilbert space. PROOF. Isometric isomorphisms preserve completeness and ~ 2 is complete. For each x s X, x s B(x ) for exactly one value of n. This fact n makes it possible to define a function in C(X ,K) by specifying its r values on B(x ) for n = 0, 1, ••• n 2 ' . DEFINITION 7.2. For f s C(X ,K) let Of= g where, for every r f(xn+l) - f(xn-1) x s B(xn)' n = 0, 1, 2, ···, g(x) = 2\ PROPOSITION 7.6. D is an operator on E. PROOF. Of = g is constant on B(x ) for each n so g E: C(X ,K) n r Since f(x ) = f(x_ ), g(x ) = 0. by proposition 7.1. 1 1 0 Also, g(x_n) = g(-xn) = g(xn) so 00 00 2 2 00 2 2 L I g ( xn) 1 = I g ( x ) I + 2 L I g ( xn) I = 2 L I g ( xn) I • Now, n=-oo 0 n=l n=l 47 :: 4~2~ jl lf(xn+l)l2 + 2CL lf(xn+l)l2t(Lif(xn-1)12t + jl If(xn-1) 12 I where the last inequality follows from Holder•s inequality for series. Each term in the last expression is finite since f E E so it follows that g E E. Thus 0 E -+ E. Also, and O(af) = aOf so 0 is linear. To show that 0 is continuous let {fk}oo be a sequence in E such k=l 48 by Holder's inequality. Now, of the last expression goes to zero as k + oo. Thus fk + 0 => Dfk + 0 so D is continuous. PROPOSITION 7.7. P = -i~D is a self-adjoint operator on E. PROOF. It suffices to show that (Pf,f) is real for every f s E. 49 so (Pf,f) * = (Pf,f) hence (Pf,f) is real. 1 Let q1 (x) = :\ [g~x)J. Then, for each real number p , the function l)J(p' ,q (x)) = g(p' (x) Sin-1 is constant on B(xn) for 1 )exprq~ (''t )j n = 0, 1, 2, ••• therefore 1J;(p 1 ,qtx)) s C(Xr,K). PROPOSITION 7.8. For each real number P1 the function ~(p 1 ,q ) 1 is a solution of the equation P~ = p 1 ~. 1 1 PROOF. Let 8 = Sin- (:\i ) • Then P1)J(p' ,q1) = - ~~ g(p') / exp r q1 (~n+1) el - exprq1 (~n-1) e II · q (xn+ ) = A 1 = A = (n+1)A = q (xn) + A and 1 1 [q(x~+ ) ] [(n+~)A J 1 q1(xn-1) = A [ q(x~-1) ] = A [(n~1)A J = (n-1)A = q1(xn) -A so P1)J(p· .q ) = _ g(p·) exp r q1 l exp{ie} _ exp{-ie} 1 ~~ (~n)e 1 1 - i1i 2 1• s1n• 8 ljJ ( p I ,q ) - _ p 1 ~ ( p 1 ,q ) . -- 25: 1 1 1 As in [3], the function g(p ) is determined from the conditions 00 1 1 2) J g*(p) exp (-im Sin- (A~ )}g(p) exp(in Sin- (?ff)]ctp = om,n -00 which are equations (Bli-a) and (B11-b) of [3]. 50 VIII. SUMMARY, CONCLUSIONS AND FURTHER PROBLEMS The main results obtained in this work are as follows. 1) The elementary length topologies determined \~ith the aid of different norms are always different (theorem 3.2). 2) The 11 measuring devices 11 norms, pseudo-norms and semi-norms over Tikhonov semi-fields may be used to give a 11 natural 11 construction of elementary length topologies on T2 topological linear spaces. 3) In almost all cases, (some exceptions are given in theorem 5.14) different 11 measuring devices•• give rise to different elementary length topologies . 4) The set C(Xr,K), as defined in chapter VII, has sufficient mathematical structure to allow a quantum mechanical analysis based on the assumption of the existence of an elementary length. It is this author•s opinion that further research involving the elementary length concept should be directed toward the development of a detailed theory involving the constant A. One attempt that might prove to be worthwhile would be to solve the Schroedinger equation defined in terms of the operator D defined in chapter VII . Recent papers by Cadzow [12] and Logan [13], [14], [15] seem to indicate that some of the techniques needed for a variational approach to the development of a physical theory involving elementary length have already been explored to some extent. The extent to which the 51 variational approach has been successful in existing physical theories suggests that a development using variational techniques would be worthy of consideration. 52 REFERENCES 1. Gudder, S. Elementary length topologies in physics, SIAM J. Appl. Math. 5(1968), 1011-1018. 2. Gamow, G. Thirty Years that Shook Physics, Doubleday, New York, 1966. 3. Atkinson, D. and Halpern, M. Non-usual topologies on space time and high-energy scattering, J. Mathematical Phys. 8(1967), 373-387. 4. Blumenthal, R.; Ehn, D.; Faissler, W.; Joseph, P.; Lanzerotti, L.; Pipkin, F.; Stairs, D. Wide-angle electron-pair production, Phys. Rev. 144(1966), 1199-1223. 5. Taylor, A. E. Introduction to Functional Analysis, John Wiley and Sons, New York, 1958. 6. Kothe, G. Topological Vector Spaces I, Springer-Verlag, New York, 1969. 7. Antonovskii, ~ ·1. Ja .; Boltjanskii, V. G.; Sarymsakov, T. A. An outline of the theory of topological semi-fields, Russian Math. Surveys 21(1966), No. 4, 163-192. 8. Iseki, Kiyoshi. An approach to locally convex topological linear spaces, Proc. Japan Acad. 40(1964), 818-820. 9. Horvath, J. Topological Vector Spaces and Distributions, Volume I, Addison-Wesley, Reading, Mass., 1966. 10. Hicks, T. L. Personal communication. 11. Hyers, D. H. Pseudo-normed linear spaces and Abelian groups, Duke Math. Journal 5(1939), 628-634. 12. Cadzow, J. A. Discrete calculus of variations, Int. J. Control 11(1970), No. 3, 393-407. 13. Logan, J. D. A canonical formalism for systems governed by certain difference equations, Notices of the AMS, Vol. 19, No. 5 (1972), A-588. 14. Logan, J. D. First integrals in the discrete variational calculus, to appear in Aequationes Mathematical. 15. Logan, J. D. Higher dimensional problems in the discrete calculus of variations, to appear in International Journal of Control. 53 VITA Jackie Ray Hamm was born on September 18, 1939, at Lurton, Arkansas. He graduated from Mt. Judea High School at Mt. Judea, Arkansas, in May 1959. In May of 1964 he received a Bachelor of Science degree from Arkansas Tech, Russellville, Arkansas, with majors in Mathematics and Physics. During the years 1964-1966 he taught High School Mathematics and Physics in Texas. From September 1966 to May 1972 he served as graduate assistant at the University of t~issouri at Rolla. He received a t1aster of Science degree in Applied Mathematics from the University of Missouri at Rolla in May of 1968. He is married to the former Charlene Nichols and they have two children, Brent and Scott.