Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$

Petr Hajek; Michal Johanis; Th. Schlumprecht

Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$

Petr Hajek, Michal Johanis, Th. Schlumprecht

Abstract

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(\Ga)$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(μ)$, $1\le p\le2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(μ)$, $1\le p<\infty$, bi-Lipschitz embeds into $c_0(\Ga)$ if and only if $\dens X<\om_\om$.

Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$

Abstract

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into

and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces

) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that

, bi-Lipschitz embeds into

if and only if

Paper Structure (20 theorems, 2 equations)

This paper contains 20 theorems, 2 equations.

Key Result

Theorem 2

A77 Every separable metric space admits a bi-Lipschitz embedding into $c_0^+$ (the non-negative cone of $c_0$).

Theorems & Definitions (35)

Definition 1
Theorem 2
Definition 3: PHK06
Definition 4: S18
Definition 5
Definition 6
Definition 7
Theorem 8: P94
proof
Lemma 12
...and 25 more