Topological classification of strong duals to nuclear (LF)-spaces

Taras Banakh

Displaying similar documents to “Topological classification of strong duals to nuclear (LF)-spaces”

Topological classification of closed convex sets in Fréchet spaces

Taras Banakh, Robert Cauty (2011)

Studia Mathematica

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We prove that each non-separable completely metrizable convex subset of a Fréchet space is homeomorphic to a Hilbert space. This resolves a more than 30 years old problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowolski and Toruńczyk, this result implies that each closed convex subset of a Fréchet space is homeomorphic to $[0, 1] ⁿ \times {[0, 1)}^{m} \times ℓ ₂ (κ)$ for some cardinals 0 ≤ n ≤ ω, 0 ≤ m ≤ 1 and κ ≥ 0.

Fréchet property in compact spaces is not preserved by $M$ -equivalence

Oleg Okunev (2005)

Commentationes Mathematicae Universitatis Carolinae

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An example of two $M$ -equivalent (hence $l$ -equivalent) compact spaces is presented, one of which is Fréchet and the other is not.

Tameness in Fréchet spaces of analytic functions

Aydın Aytuna (2016)

Studia Mathematica

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A Fréchet space with a sequence ${| | \cdot | |}_{k}_{k = 1}^{\infty}$ of generating seminorms is called tame if there exists an increasing function σ: ℕ → ℕ such that for every continuous linear operator T from into itself, there exist N₀ and C > 0 such that ${| | T (x) | | ₙ \leq C | | x | |}_{σ (n)}$ ∀x ∈ , n ≥ N₀. This property does not depend upon the choice of the fundamental system of seminorms for and is a property of the Fréchet space . In this paper we investigate tameness in the Fréchet spaces (M) of analytic functions on Stein manifolds M equipped...

Bessaga's conjecture in unstable Köthe spaces and products

Zefer Nurlu, Jasser Sarsour (1993)

Studia Mathematica

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Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases $(e_{n})$ resp. $(f_{n})$ , then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping $f_{n}$ to $t_{n} e_{π (k_{n})}$ where $(t_{n})$ is a scalar sequence, π is a permutation of ℕ and $(k_{n})$ is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods $(U_{n})$ consisting of absolutely convex sets one has ∃s ∀p ∃q...

Strong topologies on vector-valued function spaces

Marian Nowak (2000)

Czechoslovak Mathematical Journal

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Let $(X, ∥ \cdot ∥_{X})$ be a real Banach space and let $E$ be an ideal of $L^{0}$ over a $σ$ -finite measure space $(Ø, Σ, μ)$ . Let $(X)$ be the space of all strongly $Σ$ -measurable functions $f Ø \to X$ such that the scalar function $\tilde{f}$ , defined by $\tilde{f} (ø) = {∥ f (ø) ∥}_{X}$ for $ø \in Ø$ , belongs to $E$ . The paper deals with strong topologies on $E (X)$ . In particular, the strong topology $β (E (X), E {(X)}_{n}^{\sim})$ ( $E {(X)}_{n}^{\sim} =$ the order continuous dual of $E (X)$ ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

Applications of some results of infinite-dimensional topology to the topological classification of operator images

Taras Banakh, Tadeusz Dobrowolski, Anatoliĭ Plichko

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This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to $Σ^{ω}$ ” by T. Banakh. The articles summarize investigations that has been done by...

Simple construction of spaces without the Hahn-Banach extension property

Jerzy Kąkol (1992)

Commentationes Mathematicae Universitatis Carolinae

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An elementary construction for an abundance of vector topologies $ξ$ on a fixed infinite dimensional vector space $E$ such that $(E, ξ)$ has not the Hahn-Banach extension property but the topological dual ${(E, ξ)}^{'}$ separates points of $E$ from zero is given.

Closed universal subspaces of spaces of infinitely differentiable functions

Stéphane Charpentier, Quentin Menet, Augustin Mouze (2014)

Annales de l’institut Fourier

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We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a...

Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups

Außenhofer Lydia

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Abstract For a topological group G, the group G* of continuous homomorphisms (characters) into :=z∈ℂ: |z| = 1 endowed with the compact-open topology is called the character group of G and G is named ( Pontryagin) reflexive if the canonical homomorphism $α_{G} : G \to G * *$ , x ↦ (χ ↦ χ(x)), is a topological isomorphism. A comprehensive exposition of duality theory is given here. In particular, settings closely related to the theory of vector spaces (like local quasi-convexity and the corresponding hull)...