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On Realcompact Topological Vector Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector RACSAM (2011) 105:39–70 DOI 10.1007/s13398-011-0003-0 SURVEY On realcompact topological vector spaces J. K¸akol · M. López-Pellicer Received: 7 January 2010 / Accepted: 7 May 2010 / Published online: 3 February 2011 © The Author(s) 2011. This article is published with open access at Springerlink.com Abstract This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vec- tor spaces appearing in Functional Analysis, including Fréchet spaces, (LF)-spaces, and their duals, (DF)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (LF)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis. Keywords Angelicity · Baire and (b-) Baire-like · Bornological · Borel set · C∗-embedded · Class G · (DF) space · Distinguished space · Fréchet–Urysohn · k-Space · K -Analytic · (Weakly) Lindelöf (Σ) · Locally convex space · (Σ-)Quasi-Suslin space · (Strongly) realcompact space · (Compact) resolution · Talagrand compact · (Countable) tightness · Trans-separable · Weakly compact · (WCG) space · Web-bounded (compact) Mathematics Subject Classification (2000) 54H05 · 46A04 · 46A50 Dedicated to Professor Manuel Valdivia, excellent professor and mathematical researcher, on the occasion of his 80th birthday. The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation. J. K¸akol (B) Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznan, Poland e-mail: [email protected] M. López-Pellicer Departamento de Matemática Aplicada and IUMPA, Universidad Politécnica de Valencia, 46022 Valencia, Spain e-mail: [email protected] 40 J. K¸akol, M. López-Pellicer 1 Introduction For a Tichonov space (also named completely regular Hausdorff space) X by C p(X) and Cc(X) we denote the space of continuous realvalued maps on X with the pointwise and the compact-open topology, respectively. By L p(X) we denote the ∗-weak dual of C p(X).If F := { f ∈ C(X) : f (X) ⊂[0, 1]}, then in [0, 1]F the subspace {( f (x) : f ∈ F) : x ∈ X} is homeomorphic to X.WeidentifyX with this subspace {( f (x) : f ∈ F) : x ∈ X} and the closure of X in [0, 1]F is the Stone–Cechˇ compactification of X, denoted by β X.Takinginto account the restrictions to β X of the coordinate projections of [0, 1]F we deduce that each f ∈ F, and therefore each uniformly bounded f ∈ C(X) has a unique continuous extension to β X. By the realcompactification υ X of X we mean the subset of β X such that x ∈ υ X if, and only if, each f ∈ C(X) admits a continuous extension to X ∪{x}. From regularity it follows that each f ∈ C(X) admits a continuous extension to υ X. Therefore the closure in [0, 1]C(X) of {( f (x) : f ∈ C(X)) : x ∈ X} is homeomorphic to υ X. By definition X is called realcompact if X = υ X. From the continuity of the coordinate projections it follows that X is realcompact if, and only if, X is homeomorphic to a closed subspace of a cartesian product of real lines. Every metric separable space is realcompact. Clearly closed subspaces of a realcompact space are realcompact and also each product of realcompact spaces is realcompact. The intersection of a family of realcompact subspaces of a space is realcompact, because this intersection is homeomorphic to the diagonal of a product. The following well-known characterization of realcompact spaces will be used in the sequel, see [16,28]. Proposition 1 A completely regular Hausdorff space X is realcompact if, and only if, for every element x ∈ β X \X there exists h ∈ C(β X),h(X) ⊂]0, 1], i.e. which is positive on X and h(x) = 0. Proof Assume that the condition holds. Then = { −1] , ]: ∈ β \ }. X h y 0 1 y X X −1] , ] β −1] , ] As each h y 0 1 is a realcompact subspace of X (since h y 0 1 is homeomorphic to (β X×]0, 1]) ∩ G(h y), where G(h y) means the graph of h y), then X is also realcompact. Conversely, if X is real- compact and x0 ∈ β X\X = β X\υ X, then there exists a continuous function f : X → R which cannot be extended continuously to X ∪{x0}.From f (x) = max( f (x), 0) + min( f (x), 0) = 1 + max( f (x), 0) − (1 − min( f (x), 0)) we know that one of the functions g1(x) = 1 + max( f (x), 0) or g2(x) = 1 − min( f (x), 0) cannot be extended continuously to X ∪{x0}. So (*) there exists a continuous function On realcompact topological vector spaces 41 g : X →[1, ∞[ which cannot be extended continuously to X ∪{x0}.Leth be a continuous extension of the bounded function h := 1/g to β X.Ifh(x0) = 0, then we get a contradiction with (*). Hence h(x0) = 0. Recall that a uniform space X is called trans-separable [31,33] if every uniform cover of X has a countable subcover. Separable uniform spaces and Lindelöf uniform spaces are trans- separable; the converse is not true in general although every trans-separable pseudometric space is separable. Clearly a uniform space is trans-separable if, and only if, it is uniformly isomorphic to a subspace of a uniform product of separable pseudometric spaces. This implies that every uniform quasi-Suslin space [71, Chapter 1, Sect. 4.2] is trans-separable. Note also that trans- separable spaces enjoy good permanence properties. In particular, the class of trans-separable spaces is hereditary, productive and closed under uniform continuous images, see [56]. For a topological vector space (tvs, in brief) E the trans-separability means that E is iso- morphic to a subspace of the product of metrizable separable tvs. Thus, in particular if E is a locally convex space (lcs, in brief), then the weak dual (E ,σ(E , E)) of E is trans-separable. It is easy to see that a tvs E is trans-separable if, and only if, for every neighborhood of zero U in E there exists a countable subset N of E such that E = N + U,seefor example [30,46,57,58]. Also a tvs E is trans-separable if, and only if, for each continuous F-seminorm p on E the F-seminormed space (E, p) is separable, or the associated F-normed space E/ ker p is separable. The concept of trans-separable spaces has been used to study several problems both from analysis and topology, for example while studying the metrizability of precompact sets in uniform spaces and in the class of lcs, we refer the reader to papers [12,15,21,22,40,41,58, 64]. Pfister [57] proved the following: Proposition 2 A lcs E is trans-separable if, and only if, for every neighborhood of zero U in E its polar U ◦ is σ(E , E)-metrizable. This fact has been applied by Pfister [57] to show that precompact sets in (DF)-spaces are metrizable. Note the following link between realcompact and trans-separable space, see also related results of this of type in [32]. Proposition 3 A completely regular topological Hausdorff space is realcompact if, and only if, there exists an admissible uniformity N on X such that (X, N ) is trans-separable and complete. Proof If X is realcompact, then it is homeomorphic to a closed subset of RC(X). Then the induced uniformity in X is admissible complete and trans-separable. Conversely, if N is a trans-separable, complete admissible uniformity on X,then(X, N ) is isomorphic to a closed subspace of a product of metrizable separable (by trans-separability) uniform spaces. Therefore X is realcompact. Proposition 1 may suggest the following concept which originally has been introduced by K¸akol and Sliwa´ [39]. Definition 1 We shall say that X is strongly realcompact if for every sequence (xn)n of elements in β X \ X there exists f ∈ C(β X) which is positive on X and vanishes on some subsequence of (xn)n. 42 J. K¸akol, M. López-Pellicer Clearly every strongly realcompact space is realcompact. It is known [78,Exer.1B.4], that if X is locally compact σ -compact, then β X \ X is a zero set in β X,soX is strongly realcompact. Recall also that a subset A ⊂ X is said to be C-embedded ((C∗)-embedded) if every real-valued continuous (bounded and continuous) function on A can be extended to a continuous function on the whole space X. For strongly realcompact spaces we note the following property. The proof presented below from [39] uses an argument of Negrepontis [51] concerning [27, Theorem 2.7]. Proposition 4 If X is strongly realcompact, then every infinite subset D of β X \X contains an infinite subset S which is relatively compact in β X \ X and C∗-embedded in β X. Proof Let (xn)n be an injective sequence in D (i.e. xn = xm if n = m)andlet f : β X → [0, 1] be a continuous function which is positive on X and vanishes on a subsequence of (xn)n.Set −1 −1 S ={xn : n ∈ N}∩ f {0}, Yn ={x ∈ β X :|f (x)|≥n }, n ∈ N, and X1 = S ∪ Yn.
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