Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C}$ and coefficients $a_n \in \mathbb{C}$, has a so called Bohr strip, the largest strip in $\mathbb{C}$ on which $D$ converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals $1/2$. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.

We consider spaces of $2\pi$-periodic holomorphic functions $f$ on the upper halfplane $G$ which are bounded by a~weighted sup-norm $\sup_{w \in G} |f(w)|v(w)$. Here $v: G \rightarrow ]0, \infty[$ is a function which depends essentially only on $Im(w)$, $w \in G$, and satisfies $ \lim_{t \rightarrow 0} v(it) =0$. We give a complete isomorphic classification of such spaces and investigate composition operators and the differentiation operator between them.

We define and study Toeplitz operators $T_a$ with distributional symbols in the setting of weighted Fock spaces of entire functions on the complex plane. Sufficient conditions for boundedness and compactness are presented in terms of the symbol belonging to a weighted Sobolev space $W_\omega^{-m, \infty}$ of negative order.

A shorter proof is given for a theorem of Domański and Mastyło characterizing the pairs $(E,F)$ of Frèchet-Hilbert spaces with the property that every exact sequence $0\to F\to G\to E\to 0$ of Frèchet-Hilbert spaces splits. The results on acyclicity of inductive spectra of metrizable locally convex spaces which we use are also presented with proofs.

We establish a criterion to decide when a countable projective limit of countable inductive limits of normed spaces is bornological. We compare the conditions occurring within our criterion with well-known abstract conditions from the context of homological algebra and with conditions arising within the investigation of weighted PLB-spaces of continuous functions.

We prove some permanence results with respect to quotient spaces and to projective and injective tensor products of the Dunford-Pettis and Grothendieck properties in the setting of locally convex Hausdorff spaces.

The classes of Fréchet spaces not containing $l^1$, of Gelfand-Phillips spaces, and of dual Gelfand-Phillips spaces are characterized by (pre)compactness criteria for sets of bounded linear operators transforming bounded sets into precompact sets.

We unify several known three-space-properties in the context of Fréchet spaces by using the small ball properties for Fréchet spaces introduced by Frerick and the author [8].

In the paper the notion of strongly $\alpha(\cdot)$-$K$-paraconvex functions is introduced. It is shown that a strongly $\alpha(\cdot)$-$K$-paraconvex function defined on a convex set contained in a Banach space $X$ with values in $\mathbb{R}^n$ is: (a) Fréchet differentiable on a dense $G_{\delta}$-set provided $X$ is an Asplund space, (b) Gateaux differentiable on a dense $G_{\delta}$-set provided $X$ is separable.

The main result of the paper is that if $(I_n)$ is a sequence of closed ideals in an $F$-lattice $E$, then also $\sum_{n=1}^\infty I_n$, the set of all elements $x\in E$ of the form $x=\sum_n x_n$, where $x_n\in I_n$ for every $n$, is a closed ideal in $E$.

The question formulated in the title is answered in the case of a separable algebra. The necessary and sufficient condition in this case is that the ideal in question is not finitely (algebraically) generated. We conjecture that this result is true in the general case.

Corson's example shows that there exists a Banach space $E$ which is not weakly normal but $E$ contains a closed subspace isomorphic to the Banach space $C[0,1]$ and such that the quotient space $E/C[0,1]$ is isomorphic to the weakly compactly generated Banach space $c_{0}[0,1]$. This applies to show the following two results: (i) The Lindel\"of property is not a three-space property. (ii) The Lindel\"of $\Sigma$-property is not a three-space property. In this note using the lifting property developed by Susanne Dierolf we present a very simple argument providing also (ii), see Theorem 1. This argument used in the proof applies also to show that under Continuum Hypothesis every infinite-dimensional topological vector space $E$ which contains a dense hyperplane admits a stronger vector topology $\upsilon$ with the same topological dual and such that $(E,\upsilon)$ contains a dense non-Baire hyperplane. This partially answers a question of Saxon concerning Arias de Reyna-Valdivia-Saxon theorem. A Banach space $E$ has the Sobczyk Property if it contains an isomorphic copy of $c_0$ and every such a copy is complemented in $E$. The classical Sobczyk's theorem says that every separable Banach space has this property. We give an example of a $C(K)$-space $E$ and its subspace $Y$ isometric to $c_0$ such that $E/Y$ is isomorphic to $c_0(\Gamma)$, with $card(\Gamma )=2^{\aleph_0}$, yet $Y$ is uncomplemented in $E$. This complements Corson's example and shows that the Sobczyk Property (as well as the (WCG)-property, and the Separable Complementation Property) is not a~three-space property. In the last part we recall some facts (partially with a simpler presentation) concerning K-analytic, Lindelöf $\Sigma$ and analytic locally convex spaces. Additionally, a few remarks concerning weakly K-analytic spaces are included.

Let $(K,D)$ be a compact subset of an open set $D$ on a Stein manifold $\Omega$ of dimension $n$, $H^\infty(D)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm, and $A_{K}^{D}$ be a compact subset in the space of continuous functions $C(K)$ consisted of all restrictions of functions from the unit ball $\mathbb{B}_{H^\infty(D)}$. In 1950s Kolmogorov raised the problem of a strict asymptotics ([K1,K2,KT]) of an entropy of this class of analytic functions: $\mathcal{H}_{\varepsilon}(A_{K}^{D})\sim\tau(\ln\frac{1}{\varepsilon})^{n+1},\varepsilon\rightarrow 0,$ with a constant $\tau $. The main result of this paper, which generalizes and strengthens the Levin's and Tikhomirov's result in [LT], shows that this asymptotics is equivalent to the asymptotics for the widths (Kolmogorov diameters): $\ln d_{k}(A_{K}^{D})\sim -\sigma k^{1/n}, k\rightarrow \infty $, with the constant $\sigma =(\frac{2}{\tau( n+1)})^{1/n}$. This result makes it possible to get a positive solution of the above entropy problem by applying recent results [Z2] on the asymptotics for the widths $d_{k}(A_{K}^{D})$.