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Search: MSC category 46B20 ( Geometry and structure of normed linear spaces )

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26. CMB 1999 (vol 42 pp. 118)

Rao, T. S. S. R. K.
 Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-idealsCategories:46B20, 46E40

27. 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

28. CMB 1998 (vol 41 pp. 145)

Fry, R.
 Smooth partitions of unity on Banach spaces It is shown that if a Banach space $X$ admits a $C^k$-smooth bump function, and $X^{*}$ is Asplund, then $X$ admits $C^k$-smooth partitions of unity. Category:46B20

29. CMB 1998 (vol 41 pp. 225)

Vanderwerff, Jon
 Mazur intersection properties for compact and weakly compact convex sets Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively. Categories:46B03, 46B20, 46A55

30. CMB 1997 (vol 40 pp. 10)

Borwein, Jon; Vanderwerff, Jon
 Convex functions on Banach spaces not containing $\ell_1$ There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions. Categories:46A55, 46B20, 52A41
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