Abstract. We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that l ∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner Lp(X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is...

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {Ti}i=1 be N nonexpansive self-mappings of K with F = ∩i=1F (Ti) ̸= ∅ (here F (Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}i=1 is semi-compact. Let {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1) and {βn} ⊂ [τ, 1] for some τ ∈ (0, 1]. For arbitrary x0 ∈ K, let the sequence {x...

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T : C→ E a non-expansive nonself-mapping with F(T) = ∅. In this paper, we study the strong convergence of two sequences generated by xn+1 = αnx + (1− αn)(1/n+ 1) ∑n j=0(PT) xn and yn+...

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R, for a vectorial elliptic operator −∇·A(·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ when ε → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ in the case when Ω is either a smooth uniformly convex domain, or a con...

Answering an old problem in nonlinear theory, we show that c0 cannot be coarsely or uniformly embedded into a reflexive Banach space, but that any stable metric space can be coarsely and uniformly embedded into a reflexive space. We also show that certain quasi-reflexive spaces (such as the James space) also cannot be coarsely embedded into a reflexive space and that the unit ball of these spac...

In [6], the second author proved general metatheorems for the extraction of effective uniform bounds from ineffective existence proofs in functional analysis, more precisely from proofs in classical analysis A(:= weakly extensional Peano arithmetic WE-PA in all finite types + quantitifer-free choice + the axiom schema of dependent choice DC) extended with (variants of) abstract bounded metric s...

We study the regret of an online learner playing a multi-round game in a Banach space B against an adversary that plays a convex function at each round. We characterize the minimax regret when the adversary plays linear functions in terms of the Rademacher type of the dual of B. The cases when the adversary plays bounded and uniformly convex functions respectively are also considered. Our resul...

This paper deals with a sharp smoothing e ect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex ux. We brie y explain why degenerate uxes are related with the optimal smoothing e ect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out that generalized spaces of bounded variation BVΦ are particu...