Small subspaces of $L_p$

Richard Haydon; Edward Odell; Thomas Schlumprecht

doi:http://doi.org/10.4007/annals.2011.173.1.5

Abstract

We prove that if $X$ is a subspace of $L_p$ $(2\lt p\lt \infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the “infinite asymptotic game” played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970’s. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000’s.

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@article {P, MRKEY = {0126145}, AUTHOR = {Pe{\l}czy{ń}ski, A.}, TITLE = {Projections in certain {B}anach spaces}, JOURNAL = {Studia Math.}, FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica}, VOLUME = {19}, YEAR = {1960}, PAGES = {209--228}, ISSN = {0039-3223}, MRCLASS = {46.10}, MRNUMBER = {23 \#A3441}, MRREVIEWER = {L. Nachbin}, ZBLNUMBER = {0104.08503}, }
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@article {PR, MRKEY = {0361729}, AUTHOR = {Pe{\l}czy{ń}ski, A. and Rosenthal, H. P.}, TITLE = {Localization techniques in {$L\sp{p}$} spaces}, JOURNAL = {Studia Math.}, FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica}, VOLUME = {52}, YEAR = {1974/75}, PAGES = {263--289}, ISSN = {0039-3223}, MRCLASS = {46B05 (46E15)}, MRNUMBER = {50 \#14174}, MRREVIEWER = {H. E. Lacey}, ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER}, ZBLNUMBER = {0297.46023}, }
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@article {Ro, MRKEY = {2566964}, AUTHOR = {Rosendal, Christian}, TITLE = {Infinite asymptotic games}, JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$}, FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier}, VOLUME = {59}, YEAR = {2009}, NUMBER = {4}, PAGES = {1359--1384}, ISSN = {0373-0956}, CODEN = {AIFUA7}, MRCLASS = {46Bxx (03E15 05D10 91A44)}, MRNUMBER = {2566964}, URL = {http://aif.cedram.org/item?id=AIF_2009__59_4_1359_0}, ZBLNUMBER = {1187.46006}, }
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@article {R, MRKEY = {0271721}, AUTHOR = {Rosenthal, Haskell P.}, TITLE = {On the subspaces of {$L\sp{p}$} {$(p>2)$} spanned by sequences of independent random variables}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {8}, YEAR = {1970}, PAGES = {273--303}, ISSN = {0021-2172}, MRCLASS = {46.35}, MRNUMBER = {42 \#6602}, MRREVIEWER = {A. Pietsch}, DOI = {10.1007/BF02771562}, ZBLNUMBER = {0213.19303}, }
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@article {S1, MRKEY = {0390722}, AUTHOR = {Schechtman, G.}, TITLE = {Examples of {$\mathcal{L}_{p}$} spaces {$(1 JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {22}, YEAR = {1975}, NUMBER = {2}, PAGES = {138--147}, ISSN = {0021-2172}, MRCLASS = {46B05}, MRNUMBER = {52 \#11545}, MRREVIEWER = {A. S. Gleit}, DOI = {10.1007/BF02760162}, ZBLNUMBER = {0316.46018}, }
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@misc{S2, author={Schechtman, G.}, NOTE = {personal communication}, }
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@article {Z, MRKEY = {965758}, AUTHOR = {Zippin, M.}, TITLE = {Banach spaces with separable duals}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {310}, YEAR = {1988}, NUMBER = {1}, PAGES = {371--379}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {46B15 (46B10)}, MRNUMBER = {965758}, MRREVIEWER = {W. J. Davis}, DOI = {10.2307/2001128}, ZBLNUMBER = {0706.46015}, }

Small subspaces of $L_p$

Abstract

Authors