We introduce methods to “count” the number of spreading models generated by a sequence and prove the following dichotomy. Every basic sequence in a Banach space X has a normalized block basis (xn) which satisfies one and only one of the following two conditions: (1) (xn) determines a spreading model isometric to `p or c0 over X; (2) There exists a separable Banach space Y that contains (xn) and...

Let E be a Banach space. The concept of n-type over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E′′ be the second dual of E and fix g′′ 1 , . . . ,g′′ n ∈ E′′. The function τ : E×Rn → R, defined by letting τ(x,a1, . . . ,an) = ‖x+∑ni=1aig′′ i ‖ for all x ∈ E and all a1, . . . ,an ∈R, defines an n-type over E. Types that can be represented ...

Purpose: The main purpose of this paper is to prove a random fixed point theorem in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings. Results: The main finding of this paper is the identification of some random fixed point theorems and the relevant application with appropriate supporting examples. Conclusion: A random fixed point ...

We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented. It is a well-known result that a Banach space whose dual norm is Fréchet differentiable is reflexive. Also if the the third dual norm is Gâteaux differentiable the space is reflexive. For these results see e.g. [2], p.33. Similarly, by the result of [9], if the sec...

We study the extension of linear operators with range in a C(K)space, comparing and contrasting our results with the corresponding results for the nonlinear problem of extending Lipschitz maps with values in a C(K)space. We give necessary and sufficient conditions on a separable Banach space X which ensure that every operator T : E → C(K) defined on a subspace may be extended to an operator e T...

The main purpose of this paper is to establish a new variant the Hildebrandt?s theorem for Weyl spectrum in separable Banach space. This asserts that convex hull an operator T equal intersection numerical spectra operators are similar T.

Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with nonempty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lips...