### A Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2.

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We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2.

We give a characterization of $K$-weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions.

It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant $\lambda \left(v\right)=su{p}_{X}\lambda (v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in ${L}^{\infty}\left(\nu \right)$ and isometric to v and a projection ${P}_{\infty}$ from C ⊕ V onto V such that $\parallel {P}_{\infty}\parallel =\parallel P\u2081\parallel $, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P\u2081={\sum}_{i=1}^{2}{U}_{i}\otimes {v}_{i}$, then ${P}_{\infty}={\sum}_{i=1}^{2}{u}_{i}\otimes {V}_{i}$, where $d{V}_{i}=2{v}_{i}d\nu $ and $d{U}_{i}=-2{u}_{i}d\nu $.

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:{L}_{q}\left(\mu \right)\to {L}_{p}\left(\nu \right)$, each n ∈ ℕ and functions ${f}_{1},...,{f}_{n}\in {L}_{q}\left(\mu \right)$, $\left(\u0283\right({\sum}_{k=1}^{n}|T{f}_{k}{|}^{r}{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(\u0283\right({\sum}_{k=1}^{n}|{f}_{k}{|}^{r}{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.

Bourgain’s discretization theorem asserts that there exists a universal constant $C\in (0,\infty )$ with the following property. Let $X,Y$ be Banach spaces with $dimX=n$. Fix $D\in (1,\infty )$ and set $\delta ={e}^{-{n}^{Cn}}$. Assume that $\mathcal{N}$ is a $\delta $-net in the unit ball of $X$ and that $\mathcal{N}$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem.We also obtain an improvement...

It is known that $\mathcal{B}\left({\ell}^{p}\right)$ is not amenable for p = 1,2,∞, but whether or not $\mathcal{B}\left({\ell}^{p}\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $\mathcal{B}\left({\ell}^{p}\right)$ is amenable for p ∈ (1,∞), then so are ${\ell}^{\infty}\left(\mathcal{B}\left({\ell}^{p}\right)\right)$ and ${\ell}^{\infty}\left(\left({\ell}^{p}\right)\right)$. Moreover, if ${\ell}^{\infty}\left(\left({\ell}^{p}\right)\right)$ is amenable so is ${\ell}^{\infty}(,\left(E\right))$ for any index set and for any infinite-dimensional ${\mathcal{L}}^{p}$-space E; in particular, if ${\ell}^{\infty}\left(\left({\ell}^{p}\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell}^{\infty}\left(\left({\ell}^{p}\oplus \ell \xb2\right)\right)$. We show that ${\ell}^{\infty}\left(\left({\ell}^{p}\oplus \ell \xb2\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over ℕ, we exhibit...

We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${\left({\sum}_{k}{(\left(\parallel T{x}_{k}{\parallel}_{F}\right)/\left(\surd log(k+1)\right))}^{q}\right)}^{1/q}\le c\parallel {\sum}_{k}{\varepsilon}_{k}{x}_{k}{\parallel}_{{L}_{2}\left(C\left(K\right)\right)}$, for all sequences ${\left({x}_{k}\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_{k}\parallel \right)}_{k=1}^{n}$ decreasing. (2) T is of Rademacher cotype q if and only if $({\sum}_{k}{\left(\parallel T{x}_{k}{\parallel}_{F}\surd \left({\left(log(k+1)\right)}^{q}\right)\right)}^{1/q}\le c\parallel {\sum}_{k}{g}_{k}{x}_{k}{\parallel}_{{L}_{2}\left(C\left(K\right)\right)}$, for all sequences ${\left({x}_{k}\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_{k}\parallel \right)}_{k=1}^{n}$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

We study the presence of copies of ${l}_{p}^{n}$’s uniformly in the spaces ${\Pi}_{2}(C[0,1],X)$ and ${\Pi}_{1}(C[0,1],X)$. By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then ${\Pi}_{2}(C[0,1],X)$ contains $\lambda \sqrt{2}$-uniformly copies of ${l}_{\infty}^{n}$’s and ${\Pi}_{1}(C[0,1],X)$ contains $\lambda $-uniformly copies of ${l}_{2}^{n}$’s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces ${\Pi}_{2}(C[0,1],X)$ and ${\Pi}_{1}(C[0,1],X)$ are distinct, extending the well-known result that the spaces ${\Pi}_{2}(C[0,1],X)$ and $\mathcal{N}\left(C\right[0,1],X)$ are distinct.

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. ...

We compute the completely bounded Banach-Mazur distance between different finite-dimensional homogeneous Hilbertian operator spaces.

It is known that each bounded operator from lp → lris compact. The purpose of this paper is to present a very simple proof of this useful fact.