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The structure of Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2009)

Studia Mathematica

Lindenstrauss-Pełczyński (for short ℒ) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces X such that every operator from a subspace of c₀ into X can be extended to the whole c₀. Here we obtain the following structure theorem: a separable Banach space X is an ℒ-space if and only if every subspace of c₀ is placed in X in a unique position, up to automorphisms of X. This, in combination with a result of Kalton [New York J. Math. 13 (2007)], provides a negative...

The symmetric tensor product of a direct sum of locally convex spaces

José Ansemil, Klaus Floret (1998)

Studia Mathematica

An explicit representation of the n-fold symmetric tensor product (equipped with a natural topology τ such as the projective, injective or inductive one) of the finite direct sum of locally convex spaces is presented. The formula for ⨂ τ , s n ( F 1 ⨁ F 2 ) gives a direct proof of a recent result of Díaz and Dineen (and generalizes it to other topologies τ) that the n-fold projective symmetric and the n-fold projective “full” tensor product of a locally convex space E are isomorphic if E is isomorphic to its square E 2 .

The universal Banach space with a K -suppression unconditional basis

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)

Commentationes Mathematicae Universitatis Carolinae

Using the technique of Fraïssé theory, for every constant K ≥ 1 , we construct a universal object 𝕌 K in the class of Banach spaces possessing a normalized K -suppression unconditional Schauder basis.

Topological tensor products of a Fréchet-Schwartz space and a Banach space

Alfredo Peris (1993)

Studia Mathematica

We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that E ⊗ ε X is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for F ⨶ π X . This solves in the negative a question of Taskinen. We also give...

Totally convex algebras

Dieter Pumplün, Helmut Röhrl (1992)

Commentationes Mathematicae Universitatis Carolinae

By definition a totally convex algebra A is a totally convex space | A | equipped with an associative multiplication, i.eȧ morphism μ : | A | ⊗ | A | ⟶ | A | of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.

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