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Corrected typo.
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Bill Johnson
Bill Johnson
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There are many characterizations of Banach spaces that embed isometrically into $L_1$. See

Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975

for the classical results. Lindenstrauss and Pelczynski gave another useful one in their "Absolutely summing operators..." paper; namely, whenever $\sum |x^*(x_n)| \le \sum |x^*(y_n)| $, then $\sum \|x_n\| \le C \sum \|y_n\| $ where $C=1$. The main reason this is useful is that if you have the condition for some $C$, then the space $C$-embeds into $L_1$.

One quite remarkable fact (due to Lindenstrauss) is that every two dimensional Banach space embeds isometrically into $L_1$.

Greg, not surprisingly, gives the convex geometry way of looking at subspaces of $\ell_1$$L_1$, while the computer scientists I know would tell you about cut metrics and try to convince you that the really interesting problem is to understand which discrete metric spaces biLipschitz embed into $L_1$.

There are many characterizations of Banach spaces that embed isometrically into $L_1$. See

Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975

for the classical results. Lindenstrauss and Pelczynski gave another useful one in their "Absolutely summing operators..." paper; namely, whenever $\sum |x^*(x_n)| \le \sum |x^*(y_n)| $, then $\sum \|x_n\| \le C \sum \|y_n\| $ where $C=1$. The main reason this is useful is that if you have the condition for some $C$, then the space $C$-embeds into $L_1$.

One quite remarkable fact (due to Lindenstrauss) is that every two dimensional Banach space embeds isometrically into $L_1$.

Greg, not surprisingly, gives the convex geometry way of looking at subspaces of $\ell_1$, while the computer scientists I know would tell you about cut metrics and try to convince you that the really interesting problem is to understand which discrete metric spaces biLipschitz embed into $L_1$.

There are many characterizations of Banach spaces that embed isometrically into $L_1$. See

Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975

for the classical results. Lindenstrauss and Pelczynski gave another useful one in their "Absolutely summing operators..." paper; namely, whenever $\sum |x^*(x_n)| \le \sum |x^*(y_n)| $, then $\sum \|x_n\| \le C \sum \|y_n\| $ where $C=1$. The main reason this is useful is that if you have the condition for some $C$, then the space $C$-embeds into $L_1$.

One quite remarkable fact (due to Lindenstrauss) is that every two dimensional Banach space embeds isometrically into $L_1$.

Greg, not surprisingly, gives the convex geometry way of looking at subspaces of $L_1$, while the computer scientists I know would tell you about cut metrics and try to convince you that the really interesting problem is to understand which discrete metric spaces biLipschitz embed into $L_1$.

Source Link
Bill Johnson
Bill Johnson
  • 31.5k
  • 5
  • 90
  • 138

There are many characterizations of Banach spaces that embed isometrically into $L_1$. See

Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975

for the classical results. Lindenstrauss and Pelczynski gave another useful one in their "Absolutely summing operators..." paper; namely, whenever $\sum |x^*(x_n)| \le \sum |x^*(y_n)| $, then $\sum \|x_n\| \le C \sum \|y_n\| $ where $C=1$. The main reason this is useful is that if you have the condition for some $C$, then the space $C$-embeds into $L_1$.

One quite remarkable fact (due to Lindenstrauss) is that every two dimensional Banach space embeds isometrically into $L_1$.

Greg, not surprisingly, gives the convex geometry way of looking at subspaces of $\ell_1$, while the computer scientists I know would tell you about cut metrics and try to convince you that the really interesting problem is to understand which discrete metric spaces biLipschitz embed into $L_1$.