Search

[EN] It is well known that a ¿-algebra ¿ of subsets of a set ¿ verifies both Nikodým property and property (G) for the Banach space ba(¿) of bounded finitely additive measures defined in ¿. A classic result of Valdivia stating that if a ¿-algebra ¿ is covered by an increasing sequence (¿n:n¿N) of subsets, there is p¿N such that ¿p is a Nikodým set for ba(¿) was complemented in Ferrando et al. (2020) proving that there exists p¿N such that ¿p is both a Nikodým and a Grothendieck set for ba(¿). Valdivia result was the first step to get that if (¿¿:¿¿N, The second author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.

[EN] Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when C-p (X) or C-k (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X)., Research supported by Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation & Universities of Spain.

A subset Y of the dual closed unit ball BE* of a Banach space E is called a Rainwater set for E if every bounded sequence of E that converges pointwise on Y converges weakly in E. In this paper, topological properties of Rainwater sets for the Banach space Cb(X) of the real-valued continuous and bounded functions defined on a completely regular space X equipped with the supremum-norm are studied. This applies to characterize the weak K-analyticity of Cb(X) in terms of certain Rainwater sets for Cb(X). Particularly, we show that Cb(X) is weakly K-analytic if and only if there exists a Rainwater set Y for Cb(X) such that Cb(X),σY is both K-analytic and angelic, where σY denotes the topology on Cb(X) of the pointwise convergence on Y. For the case when X is compact, one gets classic Talagrand's theorem. As an application we show that if X is a compact space and Y is a Gδ-dense subspace, then X is Talagrand compact, i.e., Cp(X) is K-analytic, if and only if the space (C(X),σY) is K-analytic.

[EN] The main object of this article is to consider a family of approximation operators of exponential type, which has presumably not been studied earlier due mainly to their seemingly complicated behavior. We estimate and establish a quantitative asymptotic formula in terms of the modulus of continuity with exponential growth, a Korovkin-type result for exponential functions and also a Voronovskaja-type asymptotic formula in the simultaneous approximation., The authors wish to thank the referee for a careful reading of their submission and for a number of valuable comments and suggestions which have significantly improved this paper. The second-named author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.

The concept of Σ-base of neighborhoods of the identity of a topological group G is introduced. If the index set Σ ⊆ N N is unbounded and directed (and if additionally each subset of Σ which is bounded in N N has a bound at Σ) a base { U α : α ∈ Σ } of neighborhoods of the identity of a topological group G with U β ⊆ U α whenever α ≤ β with α , β ∈ Σ is called a Σ-base (a Σ 2 -base). The case when Σ = N N has been noticed for topological vector spaces (under the name of G -base) at [2] . If X is a separable and metrizable space which is not Polish, the space C c ( X ) has a Σ-base but does not admit any G -base. A topological group which is Frechet–Urysohn is metrizable iff it has a Σ 2 -base of the identity. Under an appropriate ZFC model the space C c ( ω 1 ) has a Σ 2 -base which is not a G -base. We also prove that (i) every compact set in a topological group with a Σ 2 -base of neighborhoods of the identity is metrizable, ( i i ) a C p ( X ) space has a Σ 2 -base iff X is countable, and ( i i i ) if a space C c ( X ) has a Σ 2 -base then X is a C-Suslin space, hence C c ( X ) is angelic.

[EN] If R is a ring of subsets of a set Omega and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily Delta of R is called a Nikodym set for ba (R) if each set {mu(a) : alpha is an element of A} in ba (R) which is pointwise bounded on Delta is norm-bounded in ba (R). If the whole ring R is a Nikodym set, R is said to have property (N), which means that R satisfies the Nikodym-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodym sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) double left right arrow (wN) holds., The first and the third authors are supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.

We describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ ( E , E ′ ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) l 1 and satisfying the Heinrich density condition we prove that ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain l 1 , then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if E ′ is separable. The last part of the paper studies the question: Which spaces ( E , σ ( E , E ′ ) ) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a ( DF )-space whose strong dual E ′ is separable, then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for a Cech-complete Lindelof space X the following are equivalent: (a) X is Polish, (b) C c ( X ) is cosmic in the weak topology, (c) the weak ⁎ -dual of C c ( X ) is an ℵ 0 -space.

[EN] While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and lp. Also recent results of Argyros, Dodos, Kanellopoulos [see 1 in the paper] and Sliwa [see 64 in the paper] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica., The first three named authors were supported by Grant PROMETEO/2013/058 of the Conselleria d'Educacio, Investigacio, Cultura i Esport of Generalitat Valenciana. The second named author was also supported by GACR Project 16-34860L and RVO: 67985840.

[EN] Let E be a Frechet space, i.e. a metrizable and complete locally convex space (lcs), E '' its strong second dual with a defining sequence of seminorms parallel to center dot parallel to(n) induced by a decreasing basis of absolutely convex neighbourhoods of zero U-n, and let H subset of E be a bounded set. Let ck(H) := sup{d(cluste(E '') (phi), E) : phi is an element of H-N} be the "worst" distance of the set of weak *-cluster points in E '' of sequences in H to E, and k(H) := sup{d(h, E) : h is an element of (H) over bar} the worst distance of (H) over bar the weak *-closure in the bidual of H to E, where d means the natural metric of E ''. Let gamma(n)(H) := sup {vertical bar lim(p) lim(m) u(p) (h(m)) - lim(m) lim(p) u(p) (h(m))vertical bar : (u(p)) subset of U-n(0), (h(m)) subset of H}, provided the involved limits exist. We extend a recent result of Angosto-Cascales to Frechet spaces by showing that: If x** is an element of (H) over bar, there is a sequence (x(p))(p) in H such that d(n)(x**, y**), [ES] Se obtienen medidas que caracterizan cuantitativamente la compacidad débil en espacios de Fréchet. De estas medidas se deducen pruebas muy sencillas de resultados de compacidad en espacios de Fréchet, extendiendo resultados previos obtenidos recientemente en espacios de Banach. Además se obtienen medidas cuantitativas de compacidad en espacios C(X) con la topología puntual, estudiando la aproximación por sucesiones, así como diferentes propiedades del espacio de funciones continuas C(X) para clases importantes de espacios X., The research was supported for the first named author by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation and by Fundacion Seneca (CARM), grant 08848/PI/08, for the second named author by National Center of Science, Poland, grant no. N N201 605340 and for the second and third authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.

Using the index of Nagami we get new topological cardinal inequalities for spaces C p ( X ) . A particular case of Theorem 1 states that if L ⊆ C p ( X ) is a Lindelof Σ-space and the Nagami index Nag ( X ) of X is less or equal than the density d ( L ) of L (which holds for instance if X is a Lindelof Σ-space), then (i) there exists a completely regular Hausdorff space Y such that Nag ( Y ) ⩽ Nag ( X ) , L ⊂ C p ( Y ) and d ( L ) = d ( Y ) ; (ii) Y admits a weaker completely regular Hausdorff topology τ ′ such that w ( Y , τ ′ ) ⩽ d ( Y ) = d ( L ) . This applies, among other things, to characterize analytic sets for the weak topology of any locally convex space E in a large class G of locally convex spaces that includes (DF)-spaces and (LF)-spaces. The latter yields a result of Cascales–Orihuela about weak metrizability of weakly compact sets in spaces from the class G .

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.

The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

We use a result of Bourgain and Delbaen on extreme points in duals of separable Banach spaces to characterize separable Banach spaces containing isomorphic copies of \( \ell^1 \) in terms of extreme points. We also study the weak star closure of a bounded subset A of a separable X Banach space in X ** in terms of the existence of a sequence in A equivalent to the canonical basis of \( \ell^1 \).

We present some selected topics of the research developed by Professor Jerzy K a kol with his collaborators. After some comments on K a kol’s Ph.D. dissertation we group together the chosen contributions in three sections: Topological vector spaces, Descriptive Topology and Functional Analysis and Other aspects in K a kol’s work. For the sake of clarity the sections have been divided in subsections.

In recent years, several quantitative counterparts for several classical such as Krein-Smulyan, Eberlein-Smulyan, Grothendieck, etc. have been proved by different authors. These new versions strengthen the original theorems and lead to new problems and applications in topology and analysis. In this survey, we present several of these quantitative versions of theorems about compactness in Banach spaces with the weak topology, Frechet spaces with the weak topology and spaces of continuous functions with the pointwise convergence topology. For example if \(H\) is a subset of a Banach space \(E\), and \(w^*\) is the weak* topology in \(E^{\prime \prime }\), the index \(k(H):=\sup \{d(x^{**},E),x^{**}\in \overline{H}^{w^{*}}\}\) is zero if and only if \(H\) is relatively compact in \((E,w)\). Then \(k(H)\) measures how far is \(H\) from being relatively compact in \((E,w)\). The following inequalty \(k(co(H))\le 2 k(H)\) is a quantitative version of the Krein-Smulian theorem about the \(w\)-relative compactness of the convex hull of a weakly compact set.

For a Tychonoff space X, we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelof $Sigma$-property of X in terms of Cp(X), we extend Okunev's results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then $nu$Y is a Lindelof $Sigma$-space (respectively K-analytic) if $nu$X has this property. In the second part, applying Christensen's theorem, we extend Pelant's result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space.We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by lp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and lp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented., The research was supported for the first named author by National Center of Science, Poland, Grant No. N N201 605340, and for the first and second named authors by Generalitat Valenciana, Conselleria d'Educacio i Esport, Spain, Grant PROMETEO/2013/058.

Many properties of a Tychonoff space \(X\) have been characterized by properties of \(C_{p}(X)\) or \(C_{c}(X)\), the spaces of continuous real-valued functions on \(X\) provided with the topology of pointwise convergence or with the compact-open topology, respectively. The question of Arhangel’skii about preservation of metrizability by \(\ell _{p}\)-equivalence in the class of first countable spaces has been partially answered by Valov in the class of those first countable spaces that are Cech complete. The preservation of complete metrizability by \(\ell _{p}\)-equivalence in the class of metrizable spaces has been obtained by Baars, de Groot and Pelant. The \(\ell _{c}\)-invariance of separable metrizability and separable complete metrizability (i.e., Polish spaces) in the class of spaces of pointwise countable type has been considered very recently by K a kol, Lopez-Pellicer and Okunev. These two \(\ell _{c}\)-invariant properties were the aim of the talk given by the author in the First Meeting in Topology and Functional Analysis, dedicated to Professor Jerzy K a kol on the occasion of his sixty birthday, September 27–28, in Elche (Spain). Here it is this talk with the proofs of properties needed to obtain these two \(\ell _{c}\)-invariant properties. As additional motivation for non specialist readers three classical \(C(X)\) theorems related with \(\ell \)-equivalence’s questions are also included.

In this chapter, we show that a tvs that is a Baire space and admits a countably compact resolution is metrizable, separable and complete. We prove that a linear map T:E→F from an F-space E having a resolution {K α :α∈ℕℕ} into a tvs F is continuous if each restriction T|K α is continuous. This theorem (due to Drewnowski) was motivated by the Arias–De Reina–Valdivia–Saxon theorem about non-Baire dense hyperplanes in Banach spaces. We provide a large class of weakly analytic metrizable and separable Baire tvs that are not analytic (clearly such spaces are necessarily not locally convex).

In this chapter, we discuss Banach spaces that have a rich family of norm-one projections onto separable subspaces. One of the tools, coming from logic, is the concept of an elementary submodel.

This chapter presents unified and direct proofs of Pfister, Cascales and Orihuela and Valdivia’s theorems about metrizability of precompact sets in (LF)-spaces, (DF)-spaces and dual metric spaces, respectively. The proofs do not require the typical machinery of quasi-Suslin spaces, upper semicontinuous compact-valued maps and so on.

In this chapter, we continue the study of spaces in the class \(\mathfrak{G}\). We prove that the weak ∗ dual (E′,σ(E′,E)) of an lcs E in the class \(\mathfrak{G}\) is K-analytic if and only if (E′,σ(E′,E)) is Lindelof if and only if (E,σ(E,E′)) has countable tightness. We show that every quasibarrelled space in the class \(\mathfrak{G}\) has countable tightness both for the weak and the original topologies. This extends a classical result of Kaplansky for a metrizable lcs. Although (DF)-spaces belong to the class \(\mathfrak{G}\), concrete examples of (DF)-spaces without countable tightness are provided. On the other hand, there are many Banach spaces E for which E endowed with the weak topology is not Lindelof. We show, however (following Khurana), that every WCG Frechet space E is weakly K-analytic. An example due to Pol showing that there exists a Banach space C(X) over a compact scattered space X such that C(X) is weakly Lindelof and not WCG is presented. We show (after Amir and Lindenstrauss) that every nonseparable reflexive Banach space contains a complemented separable subspace. Several consequences are provided.

In this chapter, we continue the study of web-compact spaces. A subclass of web-compact spaces called strongly web-compact, is introduced, and a closed graph theorem for such spaces is provided. We prove that an own product of a strongly web-compact space need not be web-compact.

This chapter deals with the class of angelic spaces, introduced by Fremlin, for which several variants of compactness coincide. A remarkable paper of Orihuela introduces a large class of topological spaces X (under the name web-compact) for which the space C p (X) is angelic. Orihuela’s theorem covers many already known partial results providing Eberlein–Smulian-type results. Following Orihuela, we show that C p (X) is angelic if X is web-compact. This yields, in particular, Talagrand’s result stating that for a compact space X the space C p (X) is K-analytic if and only if C(X) is weakly K-analytic. We present some quantitative versions of Grothendieck’s characterization of the weak compactness for spaces C(X) (for compact Hausdorff spaces X) and quantitative versions of the classical Eberlein–Grothendieck and Krein–Smulian theorems.

This chapter deals with topological (vector) spaces satisfying some sequential conditions. We study Frechet–Urysohn space (i.e., spaces E such that for each A⊂E and each \(x\in\overline{A}\) there exists a sequence in A converging to x). The main result states that every sequentially complete Frechet–Urysohn lcs is a Baire space. Since every infinite-dimensional Montel (DF)-space E is nonmetrizable and sequential, the following question arises: Is every Frechet–Urysohn space in the class \(\mathfrak{G}\) metrizable?

This chapter introduces (after Cascales and Orihuela) a large class of locally convex spaces under the name the class\(\mathfrak{G}\). The class \(\mathfrak{G}\) contains among others all (LM)-spaces (hence (LF)-spaces), and dual metric spaces (hence (DF)-spaces), spaces of distributions D′(Ω) and spaces A(Ω) of real analytic functions on open Ω⊂ℝ n . We show (following Cascales and Orihuela) that every precompact set in an lcs in the class \(\mathfrak{G}\) is metrizable. This general result covers many already known theorems for (DF)-spaces, (LF)-spaces and dual metric spaces.

In this chapter, we apply the concept of tightness to study distinguished Frechet spaces. We show that a Frechet space is distinguished if and only if its strong dual has countable tightness. This approach to studying distinguished Frechet spaces leads to a rich supply of (DF)-spaces whose weak ∗ duals are quasi-Suslin but not K-analytic. The small cardinals \(\mathfrak{b}\) and \(\mathfrak{d}\) will be used to improve the analysis of Kothe’s echelon nondistinguished Frechet space λ 1(A).

In this chapter, we prove that an lcs in the class \(\mathfrak{G}\) is metrizable if and only if E is b-Baire-like if and only if E is Frechet–Urysohn. Consequently, no proper (LB)-space is Frechet–Urysohn. We prove that if a (DF)- or (LM)-space E is sequential, then E is either metrizable or Montel (DF). We distinguish a variant of the property C 3 (due to Webb), called property \(C_{3}^{-}\) (i.e., sequential closure of any vector subspace is sequentially closed), and characterize both (DF)-spaces and (LF)-spaces with the property \(C_{3}^{-}\) as being of the form M, ϕ, or M×ϕ, where M is metrizable.

In this chapter, we show that analyticity is not a three-space property. We prove, however, that a metrizable tvs E is analytic if it contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. We reprove (using Corson’s example) that the Lindelof property is not a three-space property.

In this chapter, the class of Banach spaces having the property (C) (isolated by Corson) is studied. This property provides a large subclass of Banach spaces E whose weak topology need not be Lindelof. We collect some results of Corson, Pol, Frankiewicz, Plebanek and Ryll-Nardzewski.

[EN] Corson's example shows that there exists a Banach space EE which is not weakly normal but EE contains a closed subspace isomorphic to the Banach space C[0,1]C[0,1] and such that the quotient space E/C[0,1]E/C[0,1] is isomorphic to the weakly compactly generated Banach space c0[0,1]c0[0,1]. This applies to show the following two results: (i) The Lindelöf property is not a three-space property. (ii) The Lindelöf Σ-property is not a three-space property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinite-dimensional topological vector space EE which contains a dense hyperplane admits a stronger vector topology υυ with the same topological dual and such that (E,υ)(E,υ) contains a dense non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon theorem. A Banach space EE has the Sobczyk Property if it contains an isomorphic copy of c0c0 and every such a copy is complemented in EE. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a C(K)C(K)-space EE and its subspace YY isometric to c0c0 such that E/YE/Y is isomorphic to c0(Γ)c0(Γ), with card(Γ)=2ℵ0card(Γ)=2ℵ0, yet YY is uncomplemented in EE. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)-property, and the Separable Complementation Property) is not a~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf ΣΣ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are include, The first author has been partially supported by MEC and FEDER Project MTM2008-03211. The research for the second named author was (partially) supported by Ministry of Science and Higher Education, Poland, grant no. NN201 2740 33, and for the second and third named author by the project MTM2008 - 01502 of the Spanish Ministry of Science and Innovation.

This chapter presents several classes of nonmetrizable compact spaces that correspond to well-known classes of Banach spaces with many projections. In particular, we discuss the class of Valdivia compact spaces and its subclasses: Corson and Eberlein compact spaces. We discuss a general class of compact spaces obtained by limits of continuous retractive sequences. We also introduce the notion of a retractional skeleton, dual to projectional skeletons in Banach spaces. The last section contains an overview of Eberlein compact spaces with some classical results and examples relevant to the subjects of previous chapters.

This chapter studies analytic spaces. We show that a regular space X is analytic if and only if it has a compact resolution and admits a weaker metric topology. This fact, essentially due to Talagrand, extends Choquet’s theorem (every metric K-analytic space is analytic). Several applications will be provided. We show Christensen’s theorem stating that a separable metric topological space X is a Polish space if and only if it admits a compact resolution swallowing compact sets. We also study the following general problem: When can analyticity or K-analyticity of the weak topology σ(E,E′) of a dual pair (E,E′) be lifted to stronger topologies on E compatible with the dual pair? We prove that, if X is an uncountable analytic space, the Mackey duals L μ (X) of C p (X) is weakly analytic and not analytic. The density condition, due to Heinrich, motivates us to study the analyticity of the Mackey and strong duals of (LF)-spaces. We study trans-separable spaces and show that a tvs with a resolution of precompact sets is trans-separable. This is applied to prove that precompact sets are metrizable in any uniform space whose uniformity admits a Open image in new window -basis.

This chapter deals with the K-analyticity of a topological space E and the concept of a resolution generated on E (i.e., a family of sets {K α :α∈ℕℕ} such that \(E=\bigcup_{\alpha}K_{\alpha}\) and K α ⊂K β if α≤β). Compact resolutions (i.e., resolutions {K α :α∈ℕℕ} whose members are compact sets) naturally appear in many situations in topology and functional analysis. Any K-analytic space admits a compact resolution, and for many topological spaces X the existence of such a resolution is enough for X to be K-analytic. Many of the ideas in the book are related to the concept of compact resolution. We gather some results, mostly due to Valdivia, about lcs’s admitting resolutions consisting of Banach discs and their relations with the closed graph theorems. We present Hurewicz and Alexandrov’s theorems as well as the Calbrix–Hurewicz theorem, which yields that a regular analytic space X is not σ-compact if and only if X contains a closed subset homeomorphic to ℕℕ.