Related papers: Macphail's Theorem revisited

Bohr's last problem (1952) asks whether every ordinary Dirichlet series with nonzero Lindel\"of order function $\mu$ has $\mu'(\omega_\mu{-}0)\le-1$; a negative answer would imply Lindel\"of for $\zeta$. Kahane (1989) refuted this with…

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain.…

Let $E$ be a Banach space, and $\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\mathcal B(E)$ goes back to Johnson's seminal memoir \cite{johnson} from 1972. We present the first general…

Let $V$ be a Banach space where for fixed $n$, $1<n<\dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for…

We construct a Banach space that does not contain any infinite unconditional basic sequence.

Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse…

In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$,…

Answering one problem that has its origins in quantum mechanics, we prove that for any sequence $(A_n)_{n\in\mathbb N}$ of convex nowhere dense sets in a Banach space $X$ and any sequence $(\varepsilon_n)_{n=1}^\infty$ of positive real…

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\in\omega}$ for $X$, having the unconditional basic…

We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of…

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the…

We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements…

In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let $X$ be an arbitrary infinite-dimensional Banach space, and let the real-valued…

In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive…

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for…

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump…

We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals…

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is…

We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. S\'er. Sci. Math. Astronom. Phys. 12 (1964),…

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by…