Some problems for suggested thinking in Fréchet space theory.
Jesús M. Fernández Castillo, Marilda A. Simôes (1991)
Extracta Mathematicae
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Jesús M. Fernández Castillo, Marilda A. Simôes (1991)
Extracta Mathematicae
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José Bonet, J. D. Maitland Wright (2012)
Matematički Vesnik
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S. Dierolf, P. Domański (1993)
Studia Mathematica
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Consider the following conditions. (a) Every regular LB-space is complete; (b) if an operator T between complete LB-spaces maps bounded sets into relatively compact sets, then T factorizes through a Montel LB-space; (c) for every complete LB-space E the space C (βℕ, E) is bornological. We show that (a) ⇒ (b) ⇒ (c). Moreover, we show that if E is Montel, then (c) holds. An example of an LB-space E with a strictly increasing transfinite sequence of its Mackey derivatives is given. ...
Susanne Dierolf (1993)
Collectanea Mathematica
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We exhibit a general method to show that for several classes of Fréchet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fréchet spaces, for Fréchet spaces with the density condition and also for dual Fréchet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two real different possibilities of defining...
José Bonet, Susanne Dierolf (1989)
Revista Matemática de la Universidad Complutense de Madrid
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Klaus D. Bierstedt, José Bonet (2003)
RACSAM
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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?. ...
Mohammed Yahdi (1998)
Revista Matemática Complutense
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Let C(X) be the set of all convex and continuous functions on a separable infinite dimensional Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that the subset of all convex Fréchet-differentiable functions on X, and the subset of all (not necessarily equivalent) Fréchet-differentiable norms on X, reduce every coanalytic set, in particular they are not Borel-sets.
P. Domański, L. Drewnowski (1992)
Studia Mathematica
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Fréchet spaces of strongly, weakly and weak*-continuous Fréchet space valued functions are considered. Complete solutions are given to the problems of their injectivity or embeddability as complemented subspaces in dual Fréchet spaces.
José Bonet, Susanne Dierolf (1996)
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales
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The purpose of this note is to give an example of a distinguished Fréchet space and a non-distinguished Fréchet space which have the same inductive dual. Accordingly, distinguishedness is a property which is not reflected in the inductive dual. In contrast to this example, it was known that the properties of being quasinormable or having the density condition can be characterized in terms of the inductive dual of a Fréchet space.