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Small ball properties for Fréchet spaces.

Leonhard Frerick, Alfredo Peris (2003)

RACSAM

We give characterizations of certain properties of continuous linear maps between Fréchet spaces, as well as topological properties on Fréchet spaces, in terms of generalizations of Behrends and Kadets small ball property.

Some aspects of the modern theory of Fréchet spaces.

Klaus D. Bierstedt, José Bonet (2003)

RACSAM

We survey some recent developments in the theory of Fréchet spaces and of their duals. Among other things, Section 4 contains new, direct proofs of properties of, and results on, Fréchet spaces with the density condition, and Section 5 gives an account of the modern theory of general Köthe echelon and co-echelon spaces. The final section is devoted to the developments in tensor products of Fréchet spaces since the negative solution of Grothendieck?s ?problème des topologies?.

Some examples on quasi-barrelled spaces

Manuel Valdivia (1972)

Annales de l'institut Fourier

The three following examples are given: a bornological space containing a subspace of infinite countable codimension which is not quasi-barrelled, a quasi-barrelled 𝒟 ℱ -space containing a subspace of infinite countable codimension which is not 𝒟 ℱ -space, and bornological barrelled space which is not inductive limit of Baire space.

Some properties of short exact sequences of locally convex Riesz spaces

Stojan Radenović, Zoran Kadelburg (1998)

Commentationes Mathematicae Universitatis Carolinae

We investigate the stability of some properties of locally convex Riesz spaces in connection with subspaces and quotients and also the corresponding three-space-problems. We show that in the richer structure there are more positive answers than in the category of locally convex spaces.

Sous-espaces fermés de séries universelles sur un espace de Fréchet

Quentin Menet (2011)

Studia Mathematica

We improve a result of Charpentier [Studia Math. 198 (2010)]. We prove that even on Fréchet spaces with a continuous norm, the existence of only one restrictively universal series implies the existence of a closed infinite-dimensional subspace of restrictively universal series.

Spaces of Whitney jets on self-similar sets

Dietmar Vogt (2013)

Studia Mathematica

It is shown that complemented subspaces of s, that is, nuclear Fréchet spaces with properties (DN) and (Ω), which are 'almost normwise isomorphic' to a multiple direct sum of copies of themselves are isomorphic to s. This is applied, for instance, to spaces of Whitney jets on the Cantor set or the Sierpiński triangle and gives new results and also sheds new light on known results.

Standard exact projective resolutions relative to a countable class of Fréchet spaces

P. Domański, J. Krone, D. Vogt (1997)

Studia Mathematica

We will show that for each sequence of quasinormable Fréchet spaces ( E n ) ℕ there is a Köthe space λ such that E x t 1 ( λ ( A ) , λ ( A ) = E x t 1 ( λ ( A ) , E n ) = 0 and there are exact sequences of the form . . . → λ ( A ) → λ ( A ) → λ ( A ) → λ ( A ) → E n → 0 . If, for a fixed ℕ, E n is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form 0 → λ ( A ) → λ ( A ) → E n → 0 . The result has some applications in the theory of the functor E x t 1 in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.

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