Nonlinear Weakly Sequentially Continuous Embeddings Between Banach Spaces

Abstract

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space X coarsely (resp. uniformly) embeds into a Banach space Y by a weakly sequentially continuous map, then every spreading model (en)n of a normalized weakly null sequence in X satisfieswhere δ Y is the modulus of asymptotic uniform convexity of Y . Among many other results, we obtain Banach spaces X and Y so that X coarsely (resp. uniformly) embeds into Y , but so that X cannot be mapped into Y by a weakly sequentially continuous coarse (resp. uniform) embedding.

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…(ii) If X satisfies upper p -tree estimates, then it is trivial to check that X satisfies [28, Theorem 3(2)]. 10 Hence, by the proof of (2)⇒(3) of [28,Theorem 3], it follows that X is p -AUSable, for all p ∈ (1, ∞).…”

“…Motivated by [Bra18], in Section 6, we study coarse Lipschitz embeddings into p-AUS spaces which are also weakly sequentially continuous. Recall, a map between two Banach spaces f : X → Y is weakly sequentially continuous if for all (x n ) n in X which weakly converges to x ∈ X it follows that w-lim n f (x n ) = f (x).…”

“…Recall, a map between two Banach spaces f : X → Y is weakly sequentially continuous if for all (x n ) n in X which weakly converges to x ∈ X it follows that w-lim n f (x n ) = f (x). The author studied weakly sequentially continiuous coarse embeddings into asymptotically uniformly convex spaces and showed that the concept of coarse embeddability by a map which is also weakly sequentially continuous is strictly weaker than isomorphic embeddability and strictly stronger than coarse embeddability (see [Bra18], Corollary 1.7 and Theorem 1.8). In these notes, we prove the equivalent results for weakly sequentially continuous coarse Lipschitz embeddings.…”

“…However, since ℓ 1 is a Schur space, this example is quite unsettling. Fortunately, the next result provide us with better examples and it gives a negative answer to Problem 5.4 of [Bra18]. Banach spaces X and Y are weakly sequentially homeomorphically Lipschitzly equivalent if there exists a Lipschitz isomorphism f : X → Y such that f and f −1 are weakly sequentially continuous.…”