scholarly journals On a Banach space without a weak mid-point locally uniformly rotund norm

1997 ◽  
Vol 56 (2) ◽  
pp. 193-196 ◽  
Author(s):  
Zhibao Hu ◽  
Warren B. Moors ◽  
Mark A. Smith
Keyword(s):  
Banach Space ◽  
Rotund Norm

In this paper we show that (i) l∞ does not admit an equivalent weak mid-point locally uniformly rotund norm and (ii) l∞ / c0 does not admit an equivalent rotund norm.

A Banach space admits a locally uniformly rotund norm if its dual is a Vašák space

1990 ◽  
Vol 69 (2) ◽  
pp. 214-224 ◽  
Author(s):  
Marián Fabian ◽  
Stanimir Troyanski
Keyword(s):  
Banach Space ◽  
Rotund Norm

scholarly journals On projectional resolution of identity on the duals of certain Banach spaces

1987 ◽  
Vol 35 (3) ◽  
pp. 363-371 ◽  
Author(s):  
M. Fabian
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Differentiable Function ◽  
Continuous Derivative ◽  
Continuous Linear ◽  
Resolution Of The Identity ◽  
One To One ◽  
Rotund Norm

A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gâteaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into c0 (Γ). The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that l∞ admits no continuous Gâteaux differentiable function with bounded nonempty support.

scholarly journals A renorming theorem for dual spaces

1983 ◽  
Vol 35 (3) ◽  
pp. 334-337
Author(s):  
A. C. Yorke
Keyword(s):  
Banach Space ◽  
Dense Subset ◽  
Transfinite Induction ◽  
Dual Spaces ◽  
Induction Argument ◽  
Second Dual ◽  
Rotund Norm

AbstractIf the second dual of a Banach space E is smooth at each point of a certain norm dense subset, then its first dual admits a long sequence of norm one projections, and these projections have ranges which are suitable for a transfinite induction argument. This leads to the construction of an equivalent locally uniformly rotund norm and a Markuschevich basis for E*.

DUAL DIFFERENTIATION SPACES

2019 ◽  
Vol 99 (03) ◽  
pp. 467-472
Author(s):  
WARREN B. MOORS ◽  
NEŞET ÖZKAN TAN
Keyword(s):  
Banach Space ◽  
Open Problem ◽  
Dense Subset ◽  
Studia Math ◽  
Dual Norm ◽  
Dual Differentiation ◽  
Rotund Norm ◽  
Norm Attaining

We show that if $(X,\Vert \cdot \Vert )$ is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm $\Vert \cdot \Vert ^{\ast }$ on $X^{\ast }$ is Fréchet at the points of a dense subset of $X^{\ast }$ . This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math. 241 (2018), 71–86].

scholarly journals Asplund spaces and a variant of weak uniform rotundity

2000 ◽  
Vol 61 (3) ◽  
pp. 451-454 ◽  
Author(s):  
John Giles ◽  
Jon Vanderwerff
Keyword(s):  
Banach Space ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Uniform Rotundity ◽  
Rotund Norm ◽  
Rotund Banach Space

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.

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