Topological Vector Spaces

Schaefer, Helmut

doi:10.1007/978-1-4684-9928-5

Cited by 999 publications

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“…e.g. [14]) that the statement is equivalent to the following: For every r^-continuous seminorm ||-|| on /7 there is a r^-equicontinuous set of positive functional, C + C/7 +/ ? such that ||a||< sup 1 T(a)\ for all a^d.…”

Section: Proposition (I) F^y C>n Ifandonlyiff=0 (Ii) the Mappings Dmentioning

confidence: 99%

On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Yngvason

1984

Publ. Res. Inst. Math. Sci.

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Some basic properties of the locality ideal in Borchers's tensor algebra are established. It is shown that the ideal is a prime ideal and that the corresponding quotient algebra has a faithful Hilbert space representation. A topology is determined for which the positive cone in the quotient algebra is normal, and it is shown that every w-point distribution satisfying the locality condition is a linear combination of positive functional which also satisfy that condition. §L IntroductionThe locality ideal in Borchers 9 tensor algebra [1, 2] is the two-sided ideal generated by commutators of test functions with space-like separated supports. Its importance comes from the fact that quantum fields satisfying the requirement of local commutativity can be regarded as Hilbert space representations of the tensor algebra annihilating this ideal. Equivalently, such fields define representations of the corresponding quotient algebra. Among other things it will be shown that the states on this algebra separate points, and consequently that it has a faithful Hilbert space representation. For the tensor algebra itself this was first shown in [3]. The question whether this is also true for the quotient algebra was posed in [2], but has remained unsettled till now.The present paper is a sequel to [4, 5 3 6], and we refer to these papers and also to [2] for definitions and further references. Borchers 9 tensor algebra will be denoted by ^; it is the completed tensor algebra over the Schwartz space tf (R d ). The locality ideal 3* c is generated by elements of the form f®g - §®f with /, g^ff(R d ) satisfying the condition f(x)g(y)=Q whenever (x-y) e R d is not space-like. The method used for the investigation of 3 f e and the

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Section: Proposition (I) F^y C>n Ifandonlyiff=0 (Ii) the Mappings Dmentioning

confidence: 99%

On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Yngvason

1984

Publ. Res. Inst. Math. Sci.

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“…In terms of [32], C+ is a normal cone for a topology y on C(X) iff y has a base of neighborhoods at 0 of absolutely convex, absorbent sets W with the property figeW and f-¿ h ¿ g implies he W.lf additionally y has a base of neighborhoods W with the property |/| á \g\ and g e W implies fe W, then C(X)y is said to be locally solid. According to [32, p. 234], the continuity of the map/-^ |/| implies the continuity of the remaining mappings in (b).…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Bounded continuous functions on a completely regular space

Sentilles¹

1972

Trans. Amer. Math. Soc.

119

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Abstract. Three locally convex topologies on C(X) are introduced and developed, and in particular shown to coincide with the strict topology on locally compact A1 and yield dual spaces consisting of tight, -r-additive and x where p. is a bounded regular Borel measure on X. This yields a very satisfactory relationship between the topology on X, the space C(X), a natural class of linear functionals on it, and those measures on Jfthat measure at least the sets determined by the topology on X in the usual way, the Borel sets.This kind of representation was subsequently extended to locally compact spaces : first to functionals on the space C0(X) of continuous functions vanishing at infinity, and then further, to the bounded continuous functions on X. The last result, due to R. C. Buck, demanded the use of a locally convex topology, the strict topology, rather than a norm topology. In both extensions the same satisfactory relationship between measure and topology was obtained.In this paper we begin the development of locally convex topologies for C(X) which extend this kind of representation to its last reasonable setting, completely regular Hausdorff spaces. This setting appears to be ultimate in the sense of Hewitt's example of a regular space upon which the only continuous functions are constants. Definitions and preliminaries.The actual work of integral representation of linear forms has been done by other authors, going back to Aleksandrov [1] and, following his work, by Varadarajan [39], and later Knowles [21] and more recently Kirk [20] and Moran ([24], [25]). Our work relies heavily on theirs and will not extend the representations they have obtained but will relate these works to earlier

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“…For locally convex spaces, the notations and results from [14] are used and for general topology we refer to [6]. All vector spaces are taken over K, the field of real or complex numbers (we call K the scalar field).…”

Section: Introduction and Notationsmentioning

confidence: 99%

“…In this paper we give necessary and sufficient conditions for an equicontinuous subset of F to be σ(F , F )-compact (i.e. weakly compact in F β -note, in the notations of [14], that F β is the space F with topology of uniform convergence on the σ(F, F )-bounded subsets of F and (F β ) = F ) and then apply the results to the special cases when X is a σ-Stonian or an F -space. We will also need the following lemmas:…”

Section: Introduction and Notationsmentioning

confidence: 99%