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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]

The introduction of game-based learning techniques has significantly swayed learning, motivation, and information processing in both traditional and digital learning environments. This paper studies the footprint that the implementation of ten short-duration digital escape rooms has had on the creation of an environment of positive emotions in the educational field. The digital escape rooms were created by employing the Genial.ly platform and RPG Maker MZ software. A feelings/satisfaction questionnaire has been conducted to study what emotions students have experienced, as well as the students' opinions about essential elements of digital escape rooms, to study whether positive feelings predominate in the performance of these activities. Results show a high incidence of positive emotions, and a very favorable opinion on the tools employed and the positive feelings on the acquisition of knowledge and skills. [ABSTRACT FROM AUTHOR]

Let A be an algebra of subsets of a set Ω and b a (A) the Banach space of bounded finitely additive scalar-valued measures on A endowed with the variation norm. A subset B of A is a Nikodým set for b a (A) if each countable B -pointwise bounded subset M of b a (A) is norm bounded. A subset B of A is a Grothendieck set for b a (A) if for each bounded sequence μ n n = 1 ∞ in b a (A) the B -pointwise convergence on b a (A) implies its b a (A) * -pointwise convergence on b a (A) . A subset B of an algebra A is a strong-Nikodým (Grothendieck) set for b a (A) if in each increasing covering { B n : n ∈ N } of B there exists B m which is a Nikodým (Grothendieck) set for b a (A) . The answer of the following open question for an algebra A of subsets of a set Ω , proposed by Valdivia in 2013, has not yet been found: Is it true that if A is a Nikodým set for b a (A) then A is a strong Nikodým set for b a (A) ? In this paper we surveyed some results related to this Valdivia's open question, as well as the corresponding problem for strong Grothendieck sets. The new Propositions 1 and 3 provide more simplified proofs, particularly in their application to Theorems 1 and 2, which were the main results surveyed. Moreover, the proofs of almost all other propositions are wholly or partially original. [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]

A subset B of an algebra A of subsets of a set Ω has property (N) if each B -pointwise bounded sequence of the Banach space b a (A) is bounded in b a (A) , where b a (A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [ (V H S) ] if for each bounded sequence [if for each sequence] in b a (A) the B -pointwise convergence implies its weak convergence. B has property (s N) [ (s G) or (s V H S) ] if every increasing covering { B n : n ∈ N } of B contains a set B p with property (N) [ (G) or (V H S) ], and B has property (w N) [ (w G) or (w V H S) ] if every increasing web { B n 1 n 2 ⋯ n m : n i ∈ N , 1 ≤ i ≤ m , m ∈ N } of B contains a strand { B p 1 p 2 ⋯ p m : m ∈ N } formed by elements B p 1 p 2 ⋯ p m with property (N) [ (G) or (V H S) ] for every m ∈ N . The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every σ -algebra has properties (N) , (s N) , (G) and (V H S) . Valdivia's theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every σ -algebra has property (w N) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property (w N) of a σ -algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (w W H S) if and only if B has property (w N) and A has property (G) . [ABSTRACT FROM AUTHOR]

We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a (A) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence μ n n = 1 ∞ in b a (A) which is pointwise convergent on M is weakly convergent in b a (A) , i.e., if there is μ ∈ b a A such that μ n A → μ A for every A ∈ M then μ n → μ weakly in b a (A) . A subset M of an algebra of sets A is called a Nikodým set for b a (A) if each sequence μ n n = 1 ∞ in b a (A) which is pointwise bounded on M is bounded in b a (A) . We prove that if Σ is a σ -algebra of subsets of a set Ω which is covered by an increasing sequence Σ n : n ∈ N of subsets of Σ there exists p ∈ N such that Σ p is a Grothendieck set for b a (A) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a σ -algebra Σ is covered by an increasing sequence Σ n : n ∈ N of subsets, there is p ∈ N such that Σ p is a Nikodým set for b a Σ . This also refines the Grothendieck result stating that for each σ -algebra Σ the Banach space ℓ ∞ Σ is a Grothendieck space. Some applications to classic Banach space theory are given. [ABSTRACT FROM AUTHOR]