On the distance of subspaces of 𝑙ⁿ_{𝑝} to 𝑙^{𝑘}_{𝑝}

Johnson, William B.; Schechtman, Gideon

doi:10.1090/S0002-9947-1991-0989576-2

On the distance of subspaces of $l^ n_ p$ to $l^ k_ p$
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by William B. Johnson and Gideon Schechtman PDF

Trans. Amer. Math. Soc. 324 (1991), 319-329 Request permission

Abstract:

It is proved that if $l_p^n$ is well-isomorphic to $X \oplus Y$ and $X$ either has small dimension or is a Euclidean space, then $Y$ is well-isomorphic to $l_p^k$, $k = \operatorname {dim} Y$. The proofs use new forms of the finite dimensional decomposition method. It is shown that the constant of equivalence between a normalized $K$-unconditional basic sequence in $l_p^n$ and a subsequence of the unit vector basis of $l_p^n$ is greatest, up to a constant depending on $K$, when the sequence spans a $2$-Euclidean space.

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