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We consider the large class G of locally convex spaces that includes, among others, the classes of (D F) -spaces and (L F) -spaces. For a space E in class G we have characterized that a subspace Y of (E , σ (E , E ′)) , endowed with the induced topology, is analytic if and only if Y has a σ (E , E ′) -compact resolution and is contained in a σ (E , E ′) -separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C (X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τ p of C (X) is analytic iff (C (X) , ξ) is separable and is covered by a compact resolution. [ABSTRACT FROM AUTHOR]

This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space C k X of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology τ k . [ABSTRACT FROM AUTHOR]

The centenary of the still open problem of the existence of a separable quotient of infinite dimension in a Banach space $ X $ will be in 2032. This note aims to give a quick overview of this problem, which it is known that it has a positive solution for Banach spaces with, rough speaking, 'large density character' and also with 'small density character'. Proofs of several results have been reproduced or simplified to motivate interest in solving this so-called 'Mazur separable quotient Problem'. [ABSTRACT FROM AUTHOR]

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c 0 (Ω , X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms . n ∈ N defining its topology, then c 0 (Ω , X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f : Ω → X such that for each ε > 0 and n ∈ N the set ω ∈ Ω : f (ω) n > ε is finite or empty, with the topology defined by the semi-norms f n = sup f (ω) n : ω ∈ Ω , n ∈ N . Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c 0 (Ω , X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c 0 (Ω , X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented. [ABSTRACT FROM AUTHOR]

A local convex space E is said to be distinguished if its strong dual E β ′ has the topology β (E ′ , (E β ′) ′) , i.e., if E β ′ is barrelled. The distinguished property of the local convex space C p X of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and C p -theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space C p X is distinguished if and only if any function f ∈ R X belongs to the pointwise closure of a pointwise bounded set in C X . The extensively studied distinguished properties in the injective tensor products C p X ⊗ ε E and in C p (X , E) contrasts with the few distinguished properties of injective tensor products related to the dual space L p X of C p X endowed with the weak* topology, as well as to the weak* dual of C p (X , E) . To partially fill this gap, some distinguished properties in the injective tensor product space L p X ⊗ ε E are presented and a characterization of the distinguished property of the weak* dual of C p (X , E) for wide classes of spaces X and E is provided. [ABSTRACT FROM AUTHOR]