We prove that for s < 0, s-concave measures on ℝⁿ exhibit thin-shell concentration similar to the log-concave case. This leads to a Berry-Esseen type estimate for most of their one-dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for s-concave measures.

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it.

Let ${Rₙ}_{n=1}^{\infty }$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an ${ℒ}_p$-space, then both X and A have bases. We apply these results to show that the spaces $C_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset C\left(\right)$ and $L_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset L₁\left(\right)$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

It is shown that there is a subspace $Z_q$ of ${\ell }_q$ for $1<q<2$ which is isomorphic to ${\ell }_q$ such that ${\ell }_q/Z_q$ does not have the approximation property. On the other hand, for $2<p<\infty$ there is a subspace $Y_p$ of ${\ell }_p$ such that $Y_p$ does not have the approximation property (AP) but the quotient space ${\ell }_p/Y_p$ is isomorphic to ${\ell }_p$. The result is obtained by defining random “Enflo-Davie spaces” $Y_p$ which with full probability fail AP for all $2<p\le \infty$ and have AP for all $1\le p\le 2$. For $1<p\le 2$, $Y_p$ are isomorphic to ${\ell }_p$.

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.

We give general theorems which assert that divergence and universality of certain limiting processes are generic properties. We also define the notion of algebraic genericity, and prove that these properties are algebraically generic as well. We show that universality can occur with Dirichlet series. Finally, we give a criterion for the set of common hypercyclic vectors of a family of operators to be algebraically generic.

In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to a stochastic structural control problem. An optimal operator-valued measure, considered as the structural...

In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered...

The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in ${\ell }^1$ which are interesting from the Banach space theory point of view are discussed.

The main result: the dual of separable Banach space X contains a total subspace which is not norming over any infinite-dimensional subspace of X if and only if X has a nonquasireflexive quotient space with a strictly singular quotient mapping.