We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \mathcal{F} of X^{\ast}, which separates X, there exists a countable separating subset \mathcal{F}_{0} of \mathcal{F}. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markucevic-bases, Corson property and related geometric issues are discussed.
The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former characterisation also motivates the introduction of two indices related to the Daugavet property and a short study of them.
One shows for Banach bundles in a certain class that having a second countable locally compact Hausdorff base space and separable fibers implies the separability of the Banach space of the all sections that vanish at infinity. In the reverse direction, it is proved that for a Banach bundle with locally compact Hausdorff base space the separability of the space of all the sections that vanish at infinity implies that the base space is second countable. If the base space is compact and the space of all the sections of the bundle is generated by a weakly compact subset then the base space is an E…
We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelof number in the weak topology and other related questions.
Let $X$ be a Banach space and $Y \subseteq X$ be a closed subspace. We prove that if the quotient $X/Y$ is weakly Lindelöf determined or weak Asplund, then for every $w^*$-convergent sequence $(y_n^*)_{n\in \mathbb N}$ in $Y^*$ there exist a subsequence $(y_{n_k}^*)_{k\in \mathbb N}$ and a $w^*$-convergent sequence $(x_k^*)_{k\in \mathbb N}$ in $X^*$ such that $x_k^*|_Y=y_{n_k}^*$ for all $k\in \mathbb N$. As an application we obtain that $Y$ is Grothendieck whenever $X$ is Grothendieck and $X/Y$ is reflexive, which answers a question raised by González and Kania.
We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.
In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $σ$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.
We investigate for which compactifications $γω$ of the discrete space of natural numbers $ω$, the natural copy of the Banach space $c_0$ is complemented in $C(γω)$. We show, in particular, that the separability of the remainder of $γω$ is neither sufficient nor necessary for $c_0$ being complemented in $C(γω)$ (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of $ω$ related to embeddings of the measure algebra into $P(ω)/fin$. We also prove that a Banach space $C(K)$ contains a rich family of complemented copies of $c_0$ whenev…
In the first part of our note we prove that every Weakly Lindelöf Determined (WLD) (in particular, every reflexive) non-separable Banach $X$ space contains two dense linear subspaces $Y$ and $Z$ that are not densely isomorphic. This means that there are no further dense linear subspaces $Y_0$ and $Z_0$ of $Y$ and $Z$ which are linearly isomorphic. Our main result (Theorem B) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of $\ell_2(ω_1)$ (or more generally every WLD space of …
Let $X$ and $Y$ be separable Banach spaces and $T:X\to Y$ be a bounded linear operator. We characterize the non-separability of $T^*(Y^*)$ by means of fixing properties of the operator $T$.
Let $X$ be a separable Banach space, $Y$ be a Banach space and $Λ$ be a subset of the dual group of a given compact metrizable abelian group. We prove that if $X^*$ and $Y$ have the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) then $K(X,Y)$ has the type I-$Λ$-RNP (resp. type II-$Λ$-RNP) provided $L(X,Y)=K(X,Y)$. Some corollaries are then presented as well as results conserning the separability assumption on $X$. Similar results for the NearRNP and the WeakRNP are also presented.
We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following associations between compact topological spaces and Banach spaces: a compact K induces a Banach space C(K) of real valued continuous functions on K with the supremum norm; a Banach space X induces a compact space - the dual ball with the weak* topology. We inquire on how topological invariants on K and the dual ball are linked to the sizes of biorthogonal systems a…
Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_πY$ has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_πY$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_πY$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $…
This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, we develop a new concept of generalized …
We show that for a normal locally-${\mathscr P}$ space $X$ (where ${\mathscr P}$ is a topological property subject to some mild requirements) the subset $C_{\mathscr P}(X)$ of $C_b(X)$ consisting of those elements whose support has a neighborhood with ${\mathscr P}$, is a subalgebra of $C_b(X)$ isometrically isomorphic to $C_c(Y)$ for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification $βX$ of $X$ and contains $X$ as a dense subspace. Under certain conditions, $C_{\mathscr P}(X)$ coinci…
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindelöf in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is…
Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ or the Banach space of vector valued continuous functions $C(K, C(K, C(K ..., C(K)...)$. We address the question of the existence of complemented copies of $c_0(ω_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(ω_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, …
