Every separable Banach space with $C^{\left(n\right)}$-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and $C^{\left(n\right)}$-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.

Let 1 ≤ p < 2 and let $L_p=L_p[0,1]$ be the classical $L_p$-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f\in L_p$ spans in $L_p$ a subspace isomorphic to some Orlicz sequence space $l_M$. We give precise connections between M and f and establish conditions under which the distribution of a random variable $f\in L_p$ whose independent copies span $l_M$ in $L_p$ is essentially unique.

The purpose of this paper is to continue the investigations on the homothety of unit balls and isoperimetrices in higher-dimensional Minkowski spaces for the Holmes-Thompson measure and the Busemann measure. Moreover, we show a strong relation between affine isoperimetric inequalities and Minkowski geometry by proving some new related inequalities.

Given a Young function $\Phi$, we study the existence of copies of $c_0$ and ${\ell }_{\infty }$ in ${\mathrm{c}abv}_{\Phi }(\mu ,X)$ and in ${\mathrm{c}absv}_{\Phi }(\mu ,X)$, the countably additive, $\mu$-continuous, and $X$-valued measure spaces of bounded $\Phi$-variation and bounded $\Phi$-semivariation, respectively.

Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness....

We show that every infinite-dimensional Banach space with separable dual admits an equivalent norm which is weakly locally uniformly rotund but not locally uniformly rotund.

In this note we study the topological structure of weighted James spaces J(h). In particular we prove that J(h) is isomorphic to J if and only if the weight h is bounded. We also provide a description of J(h) if the weight is a non-decreasing sequence.

A new, unified presentation of the ideal norms of factorization of operators through Banach lattices and related ideal norms is given.

Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure ${\mu }_{x**}$ on the Cantor set. We use the measures ${\mu }_{x**}:x**\in ℬ₁\left(S¹\right)$ to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if ${\mu }_{T**(x**)}:x**\in ℬ₁\left(S¹\right)$ is non-separable as a subset of $ℳ\left(2^{\mathbb{N}}\right)$.

Two of James’ three quasi-reflexive spaces, as well as the James Tree, have the uniform $w^{*}$-Opial property.

Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that $\left(\right|\left|Tⁿx\right||/\left|\right|Tⁿ\left|\right|)\notin {\ell }^q\left(\mathbb{N}\right)$, using techniques which involve the modulus of asymptotic uniform smoothness of X.

We study order convexity and concavity of quasi-Banach Lorentz spaces ${\Lambda }_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that ${\Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\Lambda }_{p,w}$ in the case when ${\Lambda }_{p,w}$ is a normable space.