While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces $c_{0}$ and $\ell _{p}$, for $1\leq p\leq \infty $. Also recent results of Argyros, Dodos, Kanellopoulos [1] and Śliwa [64] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica.

Let us denote by $\mathcal F_1^\phi(\mathbb C)$ the space of entire functions $f\in \mathcal H(\C)$ such that $\int_\mathbb C |f(z)|e^{-\phi(|z|)}dm(z)<\infty$, where $\phi:(0,\infty)\to \mathbb R^+$ is assumed to be continuous and non-decreasing. Also given a continuous non-increasing function $v:(0,\infty)\to \mathbb R^+$ and a complex Banach space $X$, we write $H^\infty_v(\mathbb C,X)$ for the space of $X$-valued entire functions $F$ such that $\sup_{z\in \C} v(z)\|F(z)\|<\infty$. We find a very general class of weights $\phi$ and $v$ for which the space of bounded operators $\mathcal L(\mathcal F_1^\phi(\mathbb C),X)$ can be identified with $H^\infty_v(\mathbb C, X)$.

Let $X$ be a locally convex Hausdorff space with topological dual $X^*$ and $m$ be a ($\sigma$-additive) $X$-valued vector measure defined on a $\sigma$-algebra. The completeness of the associated $L^1$-space of $m$ is determined by the \emph{closedness} of $m$, a concept introduced by I. Kluvánek in the early 1970's. He characterized the closedness of $m$ via the existence of a certain kind of localizable, $[0,\infty]$- valued measure $\iota$ such that every scalar measure $\langle m,x^*\rangle:E\mto \langle m(E),x^*\rangle$, for $x^*\in X^*$, satisfies $\langle m,x^*\rangle<< \iota$. The construction of $\ia$ relies on the theory of conical measures. Unfortunately, in this generality the characterization is invalid; a counterexample is exhibited. However, by restricting $\iota$ to the class of \emph{Maharam measures} and strengthening the requirement of absolute continuity to the condition that every $\langle m,x^*\rangle$, for $x^*\in X^*$, is \emph{truly continuous } with respect to $\iota$ (a notion investigated by D. Fremlin in connection with the Radon Nikodým Theorem), it is shown that an adequate characterization of the closedness of $m$ is indeed available.

Weakly compactly generated Banach spaces and their subspaces are characterized by the presence of projectional skeletons with some additional properties. We work with real spaces. However the presented statements can be extended, without much extra effort, to complex spaces.

We study the Bohr radius of the unit ball of a~complex $n$-dimensional Banach space with an $1$-unconditional basis in terms of its lattice convexity/concavity constants. As an application we give asymptotic estimates of the Bohr radius of the unit ball of the $n$-th section of Lorentz and Marcinkiewicz sequence spaces.

Let $X$ be a complex Banach space with open unit ball $B_X.$ We describe some recent work and a number of open problems related to the maximal ideal spaces of the Fréchet algebra of holomorphic functions of bounded type on $X$ and the Banach algebra of bounded holomorphic functions on $B_X$.

We provide a sufficient condition for an operator $T$ on a non-metrizable and sequentially separable topological vector space $X$ to be sequentially hypercyclic. This condition is applied to some particular examples, namely, a composition operator on the space of real analytic functions on $]0,1[$, which solves two problems of Bonet and Doma\'nski [3], and the ``snake shift'' constructed in [5] on direct sums of sequence spaces. The two examples have in common that they do not admit a densely embedded F-space $Y$ for which the operator restricted to $Y$ is continuous and hypercyclic, i.e., the hypercyclicity of these operators cannot be a consequence of the comparison principle with hypercyclic operators on F-spaces.

A classical result of Harald Bohr linked the study of convergent and bounded Dirichlet series on the right half plane with bounded holomorphic functions on the open unit ball of the space $c_0$ of complex null sequences. Our aim here is to show that many questions in Dirichlet series have very natural solutions when, following Bohr's idea, we translate these to the infinite dimensional setting. Some are new proofs and other new results obtained by using that point of view.

In this paper we consider Toepliz operators with (locally) integrable symbols acting on Bergman spaces $A^p$ ($1<p<\infty$) of the open unit disc of the complex plane. We give a characterization of compact Toeplitz operators with symbols in $L^1$ under a mild additional condition. Our result is new even in the Hilbert space setting of $A^2$, where it extends the well-known characterization of compact Toeplitz operators with bounded symbols by Stroethoff and Zheng.