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Elementary Length Topologies Constructed Using Pseudo-Norms with Values in Tikhohov Semi-Fields

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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 Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Mathematics Commons Department: Mathematics and Statistics 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.
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  • On Almost S-Topological Vector Spaces

    On Almost S-Topological Vector Spaces

    Journal of Advanced Studies in Topology 9:2 (2018), 139–146 On almost s-topological vector spaces a, a a a Madhu Ram ∗, Shallu Sharma , Sahil Billawria , Amjad Hussain aDepartment of Mathematics, University of Jammu, JK-180006, India. Abstract In this paper, we introduce the notion of almost s-topological vector spaces and present some examples and counterexamples of almost s-topological vector spaces. Some general properties of almost s-topological vector spaces are investigated. Keywords: Semi-open sets, regular open sets, almost s-topological vector spaces. 2010 MSC: 57N17, 57N99. 1. Introduction The concept of topological vector spaces was first given by Kolmogroff [4] in 1934. Since then, different researchers have been explored many interesting and useful properties of topological vector spaces. Due to nice properties, topological vector spaces earn a great importance and remain a fundamental notion in fixed point theory, operator theory, variational inequalities, etc. Nowadays the researchers not only make use of topological vector spaces in other fields for developing new notions but also stretch and extend the concept of topological vector spaces with every possible way, making this field of study a more convenient and understandable. Recently, M. Khan et al. [2] introduced the s-topological vector spaces; a generalization of topological vector spaces. In 2016, Khan and Iqbal [3] introduced the irresolute topological vector spaces which are a particular brand of s-topological vector spaces but irresolute topological vector spaces are independent of topological vector spaces. Ibrahim [1] initiated the study of α topological vector spaces. − S. Sharma et. al. [7] put forth the concept of almost pretopological vector spaces.