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Search: MSC category 46B10 ( Duality and reflexivity [See also 46A25] )

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1. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords:super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

2. CMB 2011 (vol 54 pp. 302)

Kurka, Ondřej
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces
Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$.

Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class
Categories:46B20, 54H05, 46B10

3. CMB 2009 (vol 52 pp. 28)

Choi, Changsun; Kim, Ju Myung; Lee, Keun Young
Right and Left Weak Approximation Properties in Banach Spaces
New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

Keywords:approximation property, quasi approximation property, weak approximation property
Categories:46B28, 46B10

4. CMB 2000 (vol 43 pp. 208)

Matoušková, Eva
Extensions of Continuous and Lipschitz Functions
We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

Keywords:extension, continous, Lipschitz, Banach space
Categories:54C20, 46B10

5. CMB 1998 (vol 41 pp. 279)

Acosta, María D.; Galán, Manuel Ruiz
New characterizations of the reflexivity in terms of the set of norm attaining functionals
As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space $X$ is very smooth or its bidual satisfies the $w^{\ast }$-Mazur intersection property, then either $X$ is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if $X$ has the Mazur intersection property and so, if the norm of $X$ is Fr\'{e}chet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

Categories:46B04, 46B10, 46B20

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