Fernando Albiac

Embeddability of Snowflaked Metrics with Applications to the Nonlinear Geometry of the Spaces L p and ℓ p for 0<p<∞

We study the classical spaces Lp and ℓp for the whole range 0<p<∞ from a metric viewpoint. As we go along, we look over some of the results and techniques that, together with our work in this paper, have permitted us to obtain a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with their ad hoc distances and their snowflakings. Through connections with weaker forms of embeddings that lead to basic (yet fundamental) open problems, we also set the challenging goal of understanding the dissimilarities between the well-known subspace structure and the different nonlinear geometries that coexist inside Lp and ℓp.

Banach spaces with a unique greedy basis

The purpose of this article is to undertake an in-depth study of the properties of existence and uniqueness of greedy bases in Banach spaces. We show that greedy bases fail to exist for a range of neo-classical spaces within the family of Nakano and Orlicz sequence spaces and find the first-known cases of non-trivial spaces (i.e., different from c 0 , ź 1 , and ź 2 ) with a unique greedy basis. The variety and nature of those examples evince that a complete classification of Banach spaces with a unique greedy basis cannot be expected.

Garling sequence spaces

By generalizing a construction of Garling, for each

Bayoumi quasi-differential is not different from Fréchet-differential

1\leqslant p<\infty

1-greedy renormings of Garling sequence spaces

and each normalized, nonincreasing sequence of positive numbers

The Classical Sequence Spaces

w\in c_0\setminus\ell_1

Finite Representability of ℓ p -Spaces

we exhibit an

Banach Spaces of Continuous Functions

\ell_p

Absolutely Summing Operators

-saturated, complementably homogeneous Banach space

An Introduction to Local Theory

g(w,p)