Jerzy Kąkol

Descriptive topology in selected topics of functional analysis

Preface.- 1. Overview.- 2. Elementary Facts about Baire and Baire-Type Spaces.- 3. K-analytic and quasi-Suslin Spaces.- 4. Web-Compact Spaces and Angelic Theorems.- 5. Strongly Web-Compact Spaces and a Closed Graph Theorem.- 6. Weakly Analytic Spaces.- 7. K-analytic Baire Spaces.- 8. A Three-Space Property for Analytic Spaces.- 9. K-analytic and Analytic Spaces Cp(X).- 10. Precompact sets in (LM)-Spaces and Dual Metric Spaces.- 11. Metrizability of Compact Sets in the Class G.- 12. Weakly Realcompact Locally Convex Spaces.- 13. Corsons Propery (C) and tightness.- 14. Frechet-Urysohn Spaces and Groups.- 15. Sequential Properties in the Class G.- 16. Tightness and Distinguished Frechet Spaces.- 17. Banach Spaces with Many Projections.- 18. Spaces of Continuous Functions Over Compact Lines.- 19. Compact Spaces Generated by Retractions.- 20. Complementably Universival Banach Space.- Index.

Tightness and Distinguished Fréchet 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).

Compact spaces generated by retractions

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.

A separable Fréchet space of almost universal disposition

Abstract The Gurariĭ space is the unique separable Banach space G which is of almost universal disposition for finite-dimensional Banach spaces, which means that for every e > 0 , for all finite-dimensional normed spaces E ⊆ F , for every isometric embedding e : E → G there exists an e-isometric embedding f : F → G such that f ↾ E = e . We show that G N with a special sequence of semi-norms is of almost universal disposition for finite-dimensional graded Frechet spaces. The construction relies heavily on the universal operator on the Gurariĭ space, recently constructed by Garbulinska-Wegrzyn and the third author. In addition, we consider a non-graded sequence of semi-norms on G N with which the space G N is of almost universal disposition for finite-dimensional Frechet spaces with a fixed sequence of semi-norms. In both cases, this yields in particular that G N is universal in the class of all separable Frechet spaces.

Complementably Universal Banach Spaces

We deal with complementably universal Banach spaces. Assuming the continuum hypothesis, there exists a complementably universal Banach space of density ℵ1 for the class of Banach spaces with a projectional resolution of the identity. Similar methods produce a universal preimage for the class of Valdivia compacta of weight ℵ1.

Weakly Analytic Spaces

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.

Precompact Sets in (LM)-Spaces and Dual Metric Spaces

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.

Metrizability of Compact Sets in the Class \(\mathfrak {G}\)

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.

Fréchet–Urysohn Spaces and Groups

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?

Web-Compact Spaces and Angelic Theorems

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.