More on embedding subspaces of $$L_ p$$ in $$l^ n_ r$$.(English)Zbl 0659.46021

The author carries further his own work and those of others on embeddings of finite dimensional subspaces of $$L_ p(0,1)$$ into $$\ell^ n_ r$$.
In the present paper the author takes K-embeddings: Given two normed spaces X and Y and $$1\leq K<\infty$$, X is said to K-embed into Y (denoted $$X\hookrightarrow^{K}Y)$$ if there is a one-one linear operator $T:X\to T(X)\subseteq Y\quad with\quad \| T\| \| T^{-1}\| \leq K.$ Taking X as an m-dimensional subspace of $$L_ p$$ and Y as $$\ell^ n_ r$$, K-embeddings of X into Y are studied. The main concern is to find as to how small n can be (K,r,p,m given). Various bounds are achieved. The results obtained are improvements of earlier results. Some open problems are stated and analysed at the end.
The author also acknowledges (private communication) the improvement of the results of the current paper by Bourgain, Lindenstrauss and Milman.
Reviewer: Shaligram Singh

MSC:

 46B25 Classical Banach spaces in the general theory 46B20 Geometry and structure of normed linear spaces
Full Text:

References:

 [1] Ball, K. : Isometric embedding in lp spaces . Preprint (1984). [2] Bennett, G. , Dor, L.E. , Goodman, V. , Johnson, W.B. and Newman, C.M. : On uncomplemented subspaces of Lp, 1 < p < 2 . Israel J. Math. 26 (1977) 178-187. · Zbl 0339.46022 [3] Figiel, T. , Lindenstrauss, J. and Milman, V.D. : The dimension of almost spherical sections of convex bodies . Acta Math. 139 (1977) 53-94. · Zbl 0375.52002 [4] Johnson, W.B. and Schechtman, G. : Embedding lmp into ln1 . Acta Math. 149 (1982) 71-85. · Zbl 0522.46015 [5] Lewis, D.R. : Finite dimensional subspaces of Lp . Studia Math. 63 (1978) 207-212. · Zbl 0406.46023 [6] Maurey, B. : Théorèmes de Factorisation pour les Opérateurs à Valeurs dans un Espace Lp . Astérisque Soc. Math. France # 11 (1974). · Zbl 0278.46028 [7] Maurey, B. and Pisier, G. : Séries de variables aléatoires vectorielles indépendentes et propriétés géométriques des espaces de Banach . Studia Math. 58 (1976) 45-90. · Zbl 0344.47014 [8] Milman, V.D. and Schechtman, G. : Asymptotic Theory of Finite Dimensional Normed Spaces . Lecture Notes in Math. 1200 Springer, (1986). · Zbl 0606.46013 [9] Pisier, G. : On the dimension of the lnp-subspaces of Banach spaces, for 1 \leq p < 2 . Trans. A.M.S. 276 (1983) 201-211. · Zbl 0509.46016 [10] Schechtman, G. : Fone embeddings of finite dimensional subspaces of Lp, 1 \leq p < 2, into finite dimensional normed spaces II . Longhorn Notes. Univ. of Texas (1984/85). [11] Schechtman, G. : Fine embeddings of finite dimensional subspaces of Lp, 1 \leq p < 2, into lm1 . Proc. A.M.S. 94 (1985) 617-623. · Zbl 0597.46019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.