On complemented subspaces of sums and products of Banach spaces

In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces of $X_{p}$. We characterize the complemented subspaces of $X_{p}$ which are isomorphic to $X_{p}$ by showing that such a space must contain a canonical complemented subspace isomorphic to $X_{p}.$ We also give some characterizations of complemented subspaces of $X_{p}$ isomorphic to $\ell_{p}\oplus \ell_{2}.$

This is a survey of some of the results which were obtained in the last twelve years on the non-linear geometry of Banach spaces. We focus on the contribution of the late Nigel Kalton.

In this article, we address the following question: Is it true that the spatial numerical range (SNR) $V_A(a)$ of an element $a$ in a normed algebra $(A, \|\cdot\|)$ is always convex? If the normed algebra is unital, then it is convex \cite[Theorem 3, P.16]{BoDu:71}. In non-unital case, we believe that the problem is still open and its answer seems to be negative. In search of such a normed algebra, we have proved that the SNR $V_A(a)$ is convex in several non-unital Banach a...

In this brief note we provide a simple approach to give a new proof of the well known fact that the Banach-Alaoglu theorem and the Tychonoff product theorem for compact Hausdorff spaces are equivalent.

This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $\ell_\infty/c_0$ does not have an orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form $\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is consistent that $\ell_\in...

Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected topological spaces. Results are used in the problem of synthesis of Banach spaces, and to describe omittable spaces that are defined below.

We study properties of representing and absolutely representing systems of subspaces in Banach spaces. We also present sufficient conditions for the system of subspaces to be a representing system of subspaces.

Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F.

Let $\Ps(\N)$ be the set of all finite subsets of $\N$, endowed with the product topology. A description of the compact subsets of $\Ps(\N)$ is given. Two applications of this result to Banach space theory are shown : (1) a characterization of the symmetric sequence spaces which embed into $C(\om^\om)$, and (2) a characterization, in terms of the Orlicz function $M$, of the Orlicz sequence spaces $h_M$ which embed into $C(K)$ for some countable compact Hausdorff space $K$.

We study the question for which Tychonoff spaces $X$ and locally convex spaces $E$ the space $C_p(X,E)$ of continuous $E$-valued functions on $X$ contains a complemented copy of the space $(c_0)_p=\{x\in\mathbb{R}^\omega\colon x(n)\to0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Doma\'nski and Drewnowski, proved for the case of Banach and Fr\'echet spaces $C_k(X...