Given a Banach space X, for n ∈ N and p ∈ (1,∞) we investigate the smallest constant P ∈ (0,∞) for which every f1, . . . , fn : {−1, 1} → X satisfy ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 ∂jfj(ε) ∥∥∥∥ p dμ(ε) 6 P ∫ {−1,1}n ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 δj∆fj(ε) ∥∥∥∥ p dμ(ε)dμ(δ), where μ is the uniform probability measure on the discrete hypercube {−1, 1} and {∂j}j=1 and ∆ = ∑n j=1 ∂j are the hypercube partial derivat...

Let C(X) denote the set of all non-empty closed bounded convex subsets of a normed linear space X. In 1952 Hans R̊adström described how C(X) equipped with the Hausdorff metric could be isometrically embedded in a normed lattice with the order an extension of set inclusion. We call this lattice the R̊adström of X and denote it by R(X). We: (1) outline R̊adström’s construction, (2) examine the struc...

For a Banach space X, let B(X) denote the Banach algebra of all continuous linear operators on X. First, we study the lattice of closed ideals in B(Jp), where 1 < p < ∞ and Jp is the pth James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in B(Jp). Applications of this result include the following. (i) The Brown–McCoy radical of B(X), which by ...

We study the lattice of closed (order and algebra) ideals $\mathcal L^r(X)$ when $X$ is a Banach form $(\bigoplus _i \ell _p^i)_q$ $(p\in [1,\infty ]$, $q\in )\cup \{0\} \,\&\, p\ne q)$. show that for every such&nbsp;$X$, $\