8 research outputs found
On remotality for convex sets in Banach spaces
We show that every infinite dimensional Banach space has a closed and bounded
convex set that is not remotal.Comment: 5 pages, to appear in the Journal of Approximation Theor
Weak-star quasi norm attaining operators
For Banach spaces and , a bounded linear operator is said to weak-star quasi attain its norm if the
-closure of the image by of the unit ball of intersects
the sphere of radius centred at the origin in . This notion is
inspired by the quasi-norm attainment of operators introduced and studied in
\cite{CCJM}. As a main result, we prove that the set of weak-star quasi norm
attaining operators is dense in the space of bounded linear operators
regardless of the choice of the Banach spaces, furthermore, that the
approximating operator can be chosen with additional properties. This allows us
to distinguish the properties of weak-star quasi norm attaining operators from
those of quasi norm attaining operators. It is also shown that, under certain
conditions, weak-star quasi norm attaining operators share numbers of
equivalent properties with other types of norm attaining operators, but that
there are also a number of situations in which they behave differently from the
others.Comment: 16 page
Densely ball remotal subspaces of C(K)
AbstractWe call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)**
Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces
Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding (a0,x0)∈A×X(a0,x0)∈A×X such that d(a0,x0)=inf{d(a,x):a∈A,x∈X}d(a0,x0)=inf{d(a,x):a∈A,x∈X} (resp. d(a0,x0)=sup{d(a,x):a∈A,x∈X}d(a0,x0)=sup{d(a,x):a∈A,x∈X}). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.Dirección General de Enseñanza SuperiorJunta de AntalucíaThe Sectoral Operational Programme Human Resources Developmen
-structures in vector-valued polynomial spaces
This paper is concerned with the study of -structures in spaces of
polynomials. More precisely, we discuss for and Banach spaces, whether
the class of weakly continuous on bounded sets -homogeneous polynomials,
, is an -ideal in the space of continuous
-homogeneous polynomials . We show that there is some
hope for this to happen only for a finite range of values of . We establish
sufficient conditions under which the problem has positive and negative answers
and use the obtained results to study the particular cases when and
or is a Lorentz sequence space . We extend to our
setting the notion of property introduced by Kalton which allows us to
lift -structures from the linear to the vector-valued polynomial context.
Also, when is an -ideal in we
prove a Bishop-Phelps type result for vector-valued polynomials and relate
norm-attaining polynomials with farthest points and remotal sets