Free topological vector spaces

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.

show abstract

“…Thus

E_{β}^{''}

is a

(D F)

‐space and (iv) is proved. Finally, (v) follows from Corollary 5.20 and Theorem 6.4 of .

□

…”

Section: Quasi‐(df)‐spacesmentioning

confidence: 83%

“…Thus ′′ is a ( )-space and (iv) is proved. Finally, (v) follows from Corollary 5.20 and Theorem 6.4 of [27]. □ Note that the space Δ is not an Ascoli space by Proposition 5.12 of [4].…”

Section: Example 43mentioning

confidence: 96%

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

Ferrando

Gabriyelyan

Ka̧kol

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…Thus T ≥ µ µ µ X and hence T = µ µ µ X . Finally, the definition of the topology ν ν ν X of L(X) and Proposition 5.1 of [17] imply that the family B L is a base at zero of ν ν ν X .…”

Section: Theorem 21 ([33]mentioning

confidence: 99%

“…Since U X is the universal uniformity and X is a subspace of V(X) by Theorem 2.3 of [17], for every n ∈ N, we can choose V n ∈ U X such that y − x ∈ U n for every (x, y) ∈ V n . For every n ∈ N and each x ∈ X, choose λ(n, x) > 0 such that…”

Section: Theorem 21 ([33]mentioning

confidence: 99%

Locally convex properties of free locally convex spaces

Gabriyelyan

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent:We prove that L(X) is a (DF )-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF )-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df )-space but is not a quasi-(DF )-space. If X is a µ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford-Pettis property for every Tychonoff space X. If X is a sequential µ-space (for example, metrizable), then L(X) has the sequential Dunford-Pettis property iff X is discrete.2000 Mathematics Subject Classification. Primary 46A03, 46A08; Secondary 54C35.

show abstract

“…However, the condition on ( ) to be a barrelled space is very restrictive: the space ( ) is barrelled if and only if is discrete; see Theorem 6.4 of [4]. Let and be Tychonoff spaces.…”

Section: Proposition 4 Let and Be Tychonoff Spacesmentioning

confidence: 99%

Free Subspaces of Free Locally Convex Spaces

Gabriyelyan

Morris

2018

Journal of Function Spaces

Self Cite

View full text Add to dashboard Cite

If and are Tychonoff spaces, let ( ) and ( ) be the free locally convex space over and , respectively. For general and , the question of whether ( ) can be embedded as a topological vector subspace of ( ) is difficult. The best results in the literature are that if ( ) can be embedded as a topological vector subspace of (I), where I = [0, 1], then is a countabledimensional compact metrizable space. Further, if is a finite-dimensional compact metrizable space, then ( ) can be embedded as a topological vector subspace of (I). In this paper, it is proved that ( ) can be embedded in (R) as a topological vector subspace if is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if = R , ∈ N. It is also shown that if G and denote the Cantor space and the Hilbert cube I N , respectively, then (i) ( ) is embedded in (G) if and only if is a zero-dimensional metrizable compact space; (ii) ( ) is embedded in ( ) if and only if is a metrizable compact space.

show abstract

Free topological vector spaces

Cited by 23 publications

References 24 publications

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

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

Locally convex properties of free locally convex spaces

Free Subspaces of Free Locally Convex Spaces

Contact Info

Product

Resources

About