Expand all Collapse all | Results 26 - 34 of 34 |
26. CMB 2003 (vol 46 pp. 161)
Answer to a Question of S.~Rolewicz We exhibit examples of $F$-spaces with trivial dual which are
isomorphic to its quotient by a line, thus solving a problem in
Rolewicz's {\it Metric Linear Spaces}.
Categories:46M99, 46M15, 46A16, 46B20 |
27. CMB 2002 (vol 45 pp. 232)
On Strongly Convex Indicatrices in Minkowski Geometry The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant---(Cartan) torsion. We prove the existence of non-trivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 |
28. CMB 2002 (vol 45 pp. 3)
Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications We prove that every infinite-dimensional Banach space $X$ having a
(not necessarily equivalent) real-analytic norm is real-analytic
diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an
infinite-dimensional Banach space and $F$ is a closed subspace of $X$
such that there is a real-analytic seminorm on $X$ whose set of zeros
is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus
F$ are real-analytic diffeomorphic. As an application we show the
existence of real-analytic free actions of the circle and the
$n$-torus on certain Banach spaces.
Categories:46B20, 58B99 |
29. CMB 1999 (vol 42 pp. 221)
Boundedness of the $q$-Mean-Square Operator on Vector-Valued Analytic Martingales We study boundedness properties of the $q$-mean-square operator
$S^{(q)}$ on $E$-valued analytic martingales, where $E$ is a
complex quasi-Banach space and $2 \leq q < \infty$. We establish
that a.s. finiteness of $S^{(q)}$ for every bounded $E$-valued
analytic martingale implies strong $(p,p)$-type estimates for
$S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new
characterizations (in terms of analytic and stochastic properties
of the function $S^{(q)}$) of the complex spaces $E$ that admit an
equivalent $q$-uniformly PL-convex quasi-norm. We also obtain a
vector-valued extension (and a characterization) of part of an
observation due to Bourgain and Davis concerning the
$L^p$-boundedness of the usual square-function on scalar-valued
analytic martingales.
Categories:46B20, 60G46 |
30. CMB 1999 (vol 42 pp. 118)
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, $M$-ideals Categories:46B20, 46E40 |
31. CMB 1998 (vol 41 pp. 279)
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 |
32. CMB 1998 (vol 41 pp. 145)
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 |
33. CMB 1998 (vol 41 pp. 225)
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 |
34. CMB 1997 (vol 40 pp. 10)
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 |