We prove that the topology of the additive group of the Banach space c0 is not induced by weakly almost periodic functions or, which is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. This answers, on the negative, questions of Megrelishvili [14] and Glasner and Megrelishvili [7]. We show, in contrast, that additive groups of Schwartz locall...

A Banach space has property (S) if every normalized weakly null sequence contains a subsequence equivalent to the unit vector basis of c0. We show that the equivalence constant can be chosen "uniformly", i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture th...

If X,Y are normed spaces, let B(X,Y ) be the set of all bounded linear maps X → Y . If T : X → Y is a linear map, I take it as known that T is bounded if and only if it is continuous if and only if E ⊆ X being bounded implies that T (E) ⊆ Y is bounded. I also take it as known that B(X,Y ) is a normed space with the operator norm, that if Y is a Banach space then B(X,Y ) is a Banach space, that ...

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces nor on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated to the integra...

We study atomic decompositions and their relationship with duality and reflexivity of Banach spaces. To this end, we extend the concepts of “shrinking” and “boundedly complete” Schauder basis to the atomic decomposition framework. This allows us to answer a basic duality question: when an atomic decomposition for a Banach space generates, by duality, an atomic decomposition for its dual space. ...

In this note I prove several things about compact linear operators from one Banach space to another, especially from a Banach space to itself. Some of these may things be simpler to prove for compact operators on a Hilbert space, but since often in analysis we deal with compact operators from one Banach space to another, such as from a Sobolev space to an L space, and since the proofs here are ...

In this paper, we show that a closed convex set C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonemp...

We start by reviewing some elementary Banach and Hilbert space theory. Two key results here will prove useful in studying the properties of reproducing kernel Hilbert spaces: (a) that a linear operator on a Banach space is continuous if and only if it is bounded, and (b) that all continuous linear functionals on a Banach space arise from the inner product. The latter is often termed Riesz repre...

The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general Lp-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banac...

some algorithms for nding common xed point of a family of mappings isconstructed. indeed, let c be a nonempty closed convex subset of a uniformlyconvex banach space x whose norm is gateaux dierentiable and let {tn} bea family of self-mappings on c such that the set of all common fixed pointsof {tn} is nonempty. we construct a sequence {xn} generated by the hybridmethod and also we give the ...