On topological groups with a small base and metrizability

Descriptive Topology and Functional Analysis II

2019

Self Cite

For a Tychonoff space X, denote by P the family of topological properties P of being a convergent sequence or being a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. A maximally almost periodic (M AP ) group G respects P if P(G) = P(G + ), where G + is the group G endowed with the Bohr topology. We study relations between different respecting properties from P and show that the respecting convergent sequences (=the Schur property) is the weakest one among the properties of P. We characterize respecting properties from P in wide classes of M AP topological groups including the class of metrizable M AP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group G ∧ is c0-barrelled or g-barrelled, respectively. We prove that an lqc abelian kω-group respects all properties P ∈ P. As an application of the obtained results we show that (1) the space C k (X) is a reflexive group for every separable metrizable space X, and (2) a reflexive abelian group of finite exponent is a Mackey group. P 0 := {S, C, SC, CC, PC} and P := P 0 ∪ {FB}.Let G be a maximally almost periodic (M AP ) topological group G (for all relevant definitions, see Section 2). We denote by G + the group G endowed with the Bohr topology. Following [57], a M AP group G respects a topological property P if P(G) = P(G + ).The famous Glicksberg theorem [36] states that every locally compact abelian (LCA) group respects compactness. If a M AP group G respects compactness we shall say also that G has the Glicksberg property. Trigos-Arrieta [60,61] proved that countable compactness, pseudocompactness and functional boundedness are respected by LCA groups. Banaszczyk and Martín-Peinador [9] generalized these results to all nuclear groups. Nuclear groups were introduced and thoroughly studied by Banaszczyk in [8]. The concept of Schwartz topological abelian groups is appeared in [4]. This notion generalizes the well-known notion of a Schwartz locally convex space. All nuclear groups are Schwartz groups [4]. Außenhofer [2] proved that every locally quasi-convex Schwartz group respects compactness. For a general and simple approach to the theory of properties respected by M AP topological groups see [24].Let (E, τ ) be a locally convex space (lcs for short), E ′ the dual space of E and let τ w = σ(E, E ′ ) be the weak topology on E. Set E w := (E, τ w ). An lcs E is said to have the Schur property if E and E w have the same convergent sequences, i.e., S(E) = S(E w ). Considering E as an additive 2000 Mathematics Subject Classification. Primary 22A05, 46A03; Secondary 54H11.

Section: Preliminary Resultsmentioning

confidence: 98%

Section: Preliminary Resultsmentioning

confidence: 99%

Maximally Almost Periodic Groups and Respecting Properties

Descriptive Topology and Functional Analysis II

2019

Self Cite

“…Topological groups with a G-base were considered in [20]. A lcs E is quasibarrelled (barrelled ) if every β(E , E)-bounded (σ(E , E)-bounded) set in E is equicontinuous [34].…”

Section: ])mentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Topology and its Applications

Ka̧kol

Kubzdela

et al. 2015

Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe 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) 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 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 ǎ Cech-complete Lindelöf 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.

“…Clearly, L has even a fundamental bounded sequence. But since E does not contain a subspace linearly isomorphic to ℓ 1 and

E^{'}

is not separable, the space L does not have countable

c s^{*}

‐character by Theorem 1.8 of . (2) If E is a quasibarrelled quasi‐

(D F)

‐space, then E has a countable

c s^{*}

‐network at zero.…”

Section: More About [Fundamental] Bounded Resolutions For Spaces Cpfamentioning

confidence: 99%

“…Clearly, has even a fundamental bounded sequence. But since does not contain a subspace linearly isomorphic to 1 and ′ is not separable, the space does not have countable * -character by Theorem 1.8 of [25].…”

Section: Proposition 36 Let Be a -Space Then ( ) Has A Bounded Resmentioning

confidence: 99%

Bounded sets structure of CpX and quasi‐(DF)‐spaces

Ferrando

Mathematische Nachrichten

Ka̧kol

2019

Self Cite

For wide classes of locally convex spaces, in particular, for the space Cpfalse(Xfalse) of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of (DF)‐spaces have led us to introduce quasi‐(DF)‐spaces, a class of locally convex spaces containing (DF)‐spaces that preserves subspaces, countable direct sums and countable products. Regular (LM)‐spaces as well as their strong duals are quasi‐(DF)‐spaces. Hence the space of distributions D′false(normalΩfalse) provides a concrete example of a quasi‐(DF)‐space not being a (DF)‐space. We show that Cpfalse(Xfalse) has a fundamental bounded resolution if and only if Cpfalse(Xfalse) is a quasi‐(DF)‐space if and only if the strong dual of Cpfalse(Xfalse) is a quasi‐(DF)‐space if and only if X is countable. If X is metrizable, then Ckfalse(Xfalse) is a quasi‐(DF)‐space if and only if X is a σ‐compact Polish space.