Fr�chet differentiable norms on spaces of countable dimension

Vanderwerff, Jon D.

doi:10.1007/bf01190117

Cited by 13 publications

(12 citation statements)

References 5 publications

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“…Nevertheless, it is quite surprising that for separable spaces the completeness condition is in some sense also necessary. More precisely, Vanderwerff [20] proved that every normed space with a countable algebraic basis admits a C 1 -smooth renorming. This result was pushed further to get a C ∞ -smooth renorming [10].…”

Section: Introductionmentioning

confidence: 99%

Smooth norms in dense subspaces of Banach spaces

Dantas

Hájek

Russo

2020

Journal of Mathematical Analysis and Applications

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In the first part of our paper, we show that ℓ ∞ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ℓ 1 (c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c 0 (ω 1 ) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [8]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C ∞ -smooth norm. Finally, we prove that there is a dense subspace Y of ℓ c ∞ (ω 1 ) such that every renorming of Y contains an isometric copy of c 00 (ω 1 ).

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Section: Introductionmentioning

confidence: 99%

Smooth norms in dense subspaces of Banach spaces

Dantas

Hájek

Russo

2020

Journal of Mathematical Analysis and Applications

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“…This assertion can be proved by observing that the set of dual norms which are Fréchet differentiable at f is residual. The proof of this fact is analogous to the one given in [16,26]. Nevertheless, it is unknown whether the set of norms admitting a companion is first Baire category in N(X).…”

Section: A Baire Category Remarkmentioning

confidence: 85%

Antiproximinal Norms in Banach Spaces

Borwein

Jiménez-Sevilla

Moreno

2002

Journal of Approximation Theory

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We prove that every Banach space containing a complemented copy of c 0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the convex point of continuity property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed. © 2002 Elsevier Science (USA)

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“…The second part of this paper concerns approximation by locally smooth norms. Recently, Georgiev [4] and Vanderwerff [16] proved that, given a Banach space X and a countable set F C X \ {0}, almost all (in the Baire sense) equivalent norms in X are Frechet differentiable at each point of F. Then, it is quite natural to consider the validity of this result if we replace Frechet by a higher order of smoothness. The answer is negative due to the fact that the set of Lipschitz smooth norms at a fixed point is always of first Baire category.…”

Section: Introductionmentioning

confidence: 99%

On denseness of certain norms in Banach spaces

Sevilla¹,

Moreno²

1996

Bull. Austral. Math. Soc.

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We give several results dealing with denseness of certain classes of norms with many vertex points. We prove that, in Banach spaces with the Mazur or the weak* Mazur intersection property, every ball (convex body) can be uniformly approximated by balls (convex bodies) being the closed convex hull of their strongly vertex points. We also prove that given a countable set F, every norm can be uniformly approximated by norms which are locally linear at each point of F.

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Fr�chet differentiable norms on spaces of countable dimension

Cited by 13 publications

References 5 publications

Smooth norms in dense subspaces of Banach spaces

Smooth norms in dense subspaces of Banach spaces

Antiproximinal Norms in Banach Spaces

On denseness of certain norms in Banach spaces

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