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

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.)
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