and Applied Analysis 3 Lemma 2.7 Goebel-Kirk . Let X be a Banach space. For each ε ∈ ε0 X , 2 , one has the equality δX 2 − 2δX ε 1 − ε/2. Lemma 2.8 Ullán . Let X be a Banach space. For each 0 ≤ ε2 ≤ ε1 < 2 the following inequality holds: δX ε1 − δX ε2 ≤ ε1 − ε2 / 2 − ε1 . Using these lemmas we obtain: Theorem 2.9. Let X be a Banach space which satisfies δX 1 > 0, that is, ε0 X < 1. Then X is P...

where here, and in what follows, Eε denotes the expectation with respect to uniformly chosen ε = (ε1, . . . , εn) ∈ {−1, 1}. The infimum over all constants T for which (1.1) holds is denoted by Tp(X). An important theorem of Ribe [12] states that Banach spaces which are uniformly homeomorphic must have the same isomorphic local properties. In other words, any property which remains valid under ...

In this paper we first take a detail survey of the study of the Banach-Saks property of Banach spaces and then show the Banach-Saks property of the product spaces generated by a finite number of Banach spaces having the Banach-Saks property. A more general inequality for integrals of a class of composite functions is also given by using this property.

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A 0 , A 1) such that A 0 and A 1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype...

We work in the set theory without the axiom of choice: ZF. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gâteauxdifferentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f : F → R is a linear functional such that f ≤ p|F , then there exists a...

We give a general result on the behavior of spreading models in Banach spaces which coarse Lipschitz-embed into asymptotically uniformly convex spaces. We use this result to study the uniqueness of the uniform structure in p-sums of finite-dimensional spaces for 1 < p < ∞; in particular we give some new examples of spaces with unique uniform structure.

There are traditionally many interactions between the convex geometry community and the Banach space community. In recent years, work is being done as well on problems that are related to notions and concepts from other fields. The interaction of convex geometry and Banach space theory, and also with other areas, is due to high dimensional phenomena which lie at the crossroad of convex geometry...

Let p 2 (0; 1), let v be a weight on (0; 1) and let p (v) be the classical Lorentz space, determined by the norm kfk p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. When p 2 (1; 1), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if p (v) = ? p (v), where kfk ? p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. We may thus conc...

Replacing the nested sequence of ""nite" dimensional subspaces by the nested sequence of "closed" subspaces in the classical Bernstein lethargy theorem, we obtain a version of this theorem for the space B(X; Y) of all bounded linear maps. Using this result and some properties of diagonal operators, we investigate conditions under which a suitable pair of Banach spaces form an exact Bernstein pa...

M. Gromov [8] suggested to use uniform embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [20] and G. Kasparov and G. Yu [10] have shown that this is indeed a very powerful tool. G. Yu in [20] used the condition of embeddability into a Hilbert space; G. Kasparov and G. Yu [10] used the condition of embeddability into a g...