Fine embeddings of finite dimensional subspaces of \(L_ p\), \(1\leq p<2\), into \(\ell ^ m_ 1\). (English) Zbl 0597.46019

Schechtman, Gideon

Proc. Am. Math. Soc. 94, 617-623 (1985).

This paper answers the following basic and interesting Banach space structure question. Let m be an integer, \(\epsilon >0\), and \(2>p>1\). How large must n be for all m dimensional subspaces of \(L_ p\) to \((1+\epsilon)\)-embed in \(\ell^ n_ 1\). He shows that the embedding property is satisfied if n exceeds \(\nu m(1+1/p)(\log m)^{-1}\) where \(\nu\) depends on p and \(\epsilon\). An estimate (larger) is also found for \(p=1\).

Reviewer: D.Wulbert

MSC:

46B25	Classical Banach spaces in the general theory
46B20	Geometry and structure of normed linear spaces
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

fine embeddings; \(L_ p\)-spaces; Banach space structure; embedding property