Small subspaces of $L_p$

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.

  • [A] Go to document D. J. Aldous, "Subspaces of $L^{1}$, via random measures," Trans. Amer. Math. Soc., vol. 267, iss. 2, pp. 445-463, 1981.
    @article {A, MRKEY = {626483},
      AUTHOR = {Aldous, David J.},
      TITLE = {Subspaces of {$L\sp{1}$},
      via random measures},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {267},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {445--463},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {46B25 (60G57)},
      MRNUMBER = {626483},
      MRREVIEWER = {Gilles Pisier},
      DOI = {10.2307/1998664},
      ZBLNUMBER = {0474.46007},
      }
  • [Als] D. E. Alspach, "Tensor products and independent sums of $\mathcal{L}_p$-spaces, $1<p<\infty$," Mem. Amer. Math. Soc., vol. 138, iss. 660, p. viii, 1999.
    @article {Als, MRKEY = {1469150},
      AUTHOR = {Alspach, Dale E.},
      TITLE = {Tensor products and independent sums of {$\mathcal{L}_p$}-spaces, {$1   JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {138},
      YEAR = {1999},
      NUMBER = {660},
      PAGES = {viii+77},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {46B20 (46B28 46E30 46M05)},
      MRNUMBER = {1469150},
      MRREVIEWER = {Dirk Werner},
      }
  • [AO] Go to document D. Alspach and E. Odell, "$L_p$ spaces," in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 123-159.
    @incollection {AO, MRKEY = {1863691},
      AUTHOR = {Alspach, Dale and Odell, Edward},
      TITLE = {{$L_p$} spaces},
      BOOKTITLE = {Handbook of the Geometry of {B}anach Spaces, {V}ol. {\rm I}},
      PAGES = {123--159},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2001},
      MRCLASS = {46B20 (46E30)},
      MRNUMBER = {1863691},
      MRREVIEWER = {Y. Benyamini},
      DOI = {10.1016/S1874-5849(01)80005-X},
      ZBLNUMBER = {1008.46013},
      }
  • [BRS] Go to document J. Bourgain, H. P. Rosenthal, and G. Schechtman, "An ordinal $L^{p}$-index for Banach spaces, with application to complemented subspaces of $L^{p}$," Ann. of Math., vol. 114, iss. 2, pp. 193-228, 1981.
    @article {BRS, MRKEY = {632839},
      AUTHOR = {Bourgain, J. and Rosenthal, H. P. and Schechtman, G.},
      TITLE = {An ordinal {$L\sp{p}$}-index for {B}anach spaces, with application to complemented subspaces of {$L\sp{p}$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {114},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {193--228},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {46B25 (46E30)},
      MRNUMBER = {632839},
      MRREVIEWER = {Gilles Pisier},
      DOI = {10.2307/1971293},
      ZBLNUMBER = {0496.46010},
      }
  • [B] D. L. Burkholder, "Distribution function inequalities for martingales," Ann. Probability, vol. 1, pp. 19-42, 1973.
    @article {B, MRKEY = {0365692},
      AUTHOR = {Burkholder, D. L.},
      TITLE = {Distribution function inequalities for martingales},
      JOURNAL = {Ann. Probability},
      VOLUME = {1},
      YEAR = {1973},
      PAGES = {19--42},
      MRCLASS = {60G45 (30A78)},
      MRNUMBER = {51 \#1944},
      MRREVIEWER = {Martin L. Silverstein},
      ZBLNUMBER = {0301.60035},
      }
  • [BDG] D. L. Burkholder, B. J. Davis, and R. F. Gundy, "Integral inequalities for convex functions of operators on martingales," in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, 1972, pp. 223-240.
    @inproceedings {BDG, MRKEY = {0400380},
      AUTHOR = {Burkholder, D. L. and Davis, B. J. and Gundy, R. F.},
      TITLE = {Integral inequalities for convex functions of operators on martingales},
      BOOKTITLE = {Proceedings of the {S}ixth {B}erkeley {S}ymposium on {M}athematical {S}tatistics and {P}robability},
      VENUE={{U}niv. {C}alifornia, {B}erkeley, {C}alif., 1970/1971, {V}ol. {II}: {P}robability Theory},
      PAGES = {223--240},
      PUBLISHER = {Univ. California Press},
      ADDRESS = {Berkeley, CA},
      YEAR = {1972},
      MRCLASS = {60G15},
      MRNUMBER = {53 \#4214},
      MRREVIEWER = {Maurizio Pratelli},
      ZBLNUMBER = {0253.60056},
      }
  • [EW] I. S. Èdel$’$vsteuin and P. Wojtaszczyk, "On projections and unconditional bases in direct sums of Banach spaces," Studia Math., vol. 56, iss. 3, pp. 263-276, 1976.
    @article {EW, MRKEY = {0425585},
      AUTHOR = {{È}del{$'$}{š}te{\u\i}n, I. S. and Wojtaszczyk, P.},
      TITLE = {On projections and unconditional bases in direct sums of {B}anach spaces},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      VOLUME = {56},
      YEAR = {1976},
      NUMBER = {3},
      PAGES = {263--276},
      ISSN = {0039-3223},
      MRCLASS = {46B15},
      MRNUMBER = {54 \#13539},
      MRREVIEWER = {Gideon Schechtman},
      ZBLNUMBER = {0362.46017},
      }
  • [Garling] D. J. H. Garling, "Stable Banach spaces, random measures and Orlicz function spaces," in Probability Measures on Groups, New York: Springer-Verlag, 1982, vol. 928, pp. 121-175.
    @incollection {Garling, MRKEY = {669066},
      AUTHOR = {Garling, D. J. H.},
      TITLE = {Stable {B}anach spaces, random measures and {O}rlicz function spaces},
      BOOKTITLE = {Probability Measures on Groups},
      VENUE={{O}berwolfach, 1981},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {928},
      PAGES = {121--175},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      MRCLASS = {46B20 (46E40 46G99 60B11 60G57)},
      MRNUMBER = {669066},
      MRREVIEWER = {W. A. Woyczy{ń}ski},
      ZBLNUMBER = {0485.46012},
      }
  • [GLR] Go to document Y. Gordon, D. R. Lewis, and J. R. Retherford, "Banach ideals of operators with applications," J. Functional Analysis, vol. 14, pp. 85-129, 1973.
    @article {GLR, MRKEY = {0380488},
      AUTHOR = {Gordon, Y. and Lewis, D. R. and Retherford, J. R.},
      TITLE = {Banach ideals of operators with applications},
      JOURNAL = {J. Functional Analysis},
      VOLUME = {14},
      YEAR = {1973},
      PAGES = {85--129},
      MRCLASS = {47B10 (46L20)},
      MRNUMBER = {52 \#1388},
      MRREVIEWER = {D. J. H. Garling},
      ZBLNUMBER = {0272.47024},
      DOI={10.1016/0022-1236(73)90031-1},
      }
  • [G-D] S. Guerre-Delabrière, Classical Sequences in Banach Spaces, New York: Marcel Dekker, 1992, vol. 166.
    @book {G-D, MRKEY = {1197117},
      AUTHOR = {Guerre-Delabri{è}re, Sylvie},
      TITLE = {Classical Sequences in {B}anach Spaces},
      SERIES = {Monogr. and Textbooks in Pure and Applied Math.},
      VOLUME= {166},
      NOTE = {With a foreword by Haskell P. Rosenthal},
      PUBLISHER = {Marcel Dekker},
      ADDRESS = {New York},
      YEAR = {1992},
      PAGES = {xvi+207},
      ISBN = {0-8247-8723-4},
      MRCLASS = {46B15 (46B45 46E30)},
      MRNUMBER = {1197117},
      MRREVIEWER = {Mikhail Ostrovski{\u\i}},
      ZBLNUMBER = {0756.46007},
      }
  • [Hit] P. Hitczenko, "Best constants in martingale version of Rosenthal’s inequality," Ann. Probab., vol. 18, iss. 4, pp. 1656-1668, 1990.
    @article {Hit, MRKEY = {1071816},
      AUTHOR = {Hitczenko, Pawe{\l}},
      TITLE = {Best constants in martingale version of {R}osenthal's inequality},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {18},
      YEAR = {1990},
      NUMBER = {4},
      PAGES = {1656--1668},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {60E15 (60G42)},
      MRNUMBER = {1071816},
      MRREVIEWER = {Deli Li},
      URL = {http://links.jstor.org/sici?sici=0091-1798(199010)18:4<1656:BCIMVO>2.0.CO;2-8&origin=MSN},
      ZBLNUMBER = {0725.60018},
      }
  • [J] Go to document W. B. Johnson, "On quotients of $L_{p}$ which are quotients of $l_{p}$," Compositio Math., vol. 34, iss. 1, pp. 69-89, 1977.
    @article {J, MRKEY = {0454595},
      AUTHOR = {Johnson, W. B.},
      TITLE = {On quotients of {$L_{p}$} which are quotients of {$l_{p}$}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {34},
      YEAR = {1977},
      NUMBER = {1},
      PAGES = {69--89},
      ISSN = {0010-437X},
      MRCLASS = {46B05},
      MRNUMBER = {56 \#12844},
      MRREVIEWER = {Bernd Rosenberger},
      URL = {http://www.numdam.org/item?id=CM_1977__34_1_69_0},
      ZBLNUMBER = {0375.46023},
      }
  • [JMST] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, "Symmetric structures in Banach spaces," Mem. Amer. Math. Soc., vol. 19, iss. 217, p. v, 1979.
    @article {JMST, MRKEY = {527010},
      AUTHOR = {Johnson, W. B. and Maurey, B. and Schechtman, G. and Tzafriri, L.},
      TITLE = {Symmetric structures in {B}anach spaces},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {19},
      YEAR = {1979},
      NUMBER = {217},
      PAGES = {v+298},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {46B25 (46E30 47B38)},
      MRNUMBER = {527010},
      MRREVIEWER = {Z. Chanturia},
      }
  • [JO1] W. B. Johnson and E. Odell, "Subspaces of $L_{p}$ which embed into $l_{p}$," Compositio Math., vol. 28, pp. 37-49, 1974.
    @article {JO1, MRKEY = {0352938},
      AUTHOR = {Johnson, W. B. and Odell, E.},
      TITLE = {Subspaces of {$L_{p}$} which embed into {$l_{p}$}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {28},
      YEAR = {1974},
      PAGES = {37--49},
      ISSN = {0010-437X},
      MRCLASS = {46B05},
      MRNUMBER = {50 \#5424},
      MRREVIEWER = {E. Dubinsky},
      ZBLNUMBER = {0282.46020},
      }
  • [JO2] Go to document W. B. Johnson and E. Odell, "Subspaces and quotients of $l_{p}\oplus l_{2}$ and $X_{p}$," Acta Math., vol. 147, iss. 1-2, pp. 117-147, 1981.
    @article {JO2, MRKEY = {631092},
      AUTHOR = {Johnson, W. B. and Odell, E.},
      TITLE = {Subspaces and quotients of {$l_{p}\oplus l_{2}$} and {$X_{p}$}},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {147},
      YEAR = {1981},
      NUMBER = {1-2},
      PAGES = {117--147},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {46B20 (46E30)},
      MRNUMBER = {631092},
      MRREVIEWER = {G. Schechtman},
      DOI = {10.1007/BF02392872},
      ZBLNUMBER = {0484.46020},
      }
  • [JL] Go to document W. B. Johnson and J. Lindenstrauss, "Basic concepts in the geometry of Banach spaces," in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 1-84.
    @incollection {JL, MRKEY = {1863689},
      AUTHOR = {Johnson, William B. and Lindenstrauss, Joram},
      TITLE = {Basic concepts in the geometry of {B}anach spaces},
      BOOKTITLE = {Handbook of the Geometry of {B}anach Spaces, {V}ol. {\rm I}},
      PAGES = {1--84},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2001},
      MRCLASS = {46Bxx (46-00 46-01)},
      MRNUMBER = {2003f:46013},
      MRREVIEWER = {G. J. O. Jameson},
      DOI = {10.1016/S1874-5849(01)80003-6},
      ZBLNUMBER = {1011.46009},
      }
  • [JRZ] Go to document W. B. Johnson, H. P. Rosenthal, and M. Zippin, "On bases, finite dimensional decompositions and weaker structures in Banach spaces," Israel J. Math., vol. 9, pp. 488-506, 1971.
    @article {JRZ, MRKEY = {0280983},
      AUTHOR = {Johnson, W. B. and Rosenthal, H. P. and Zippin, M.},
      TITLE = {On bases, finite dimensional decompositions and weaker structures in {B}anach spaces},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {9},
      YEAR = {1971},
      PAGES = {488--506},
      ISSN = {0021-2172},
      MRCLASS = {46.10},
      MRNUMBER = {0280983},
      MRREVIEWER = {J. T. Marti},
      DOI = {10.1007/BF02771464},
      ZBLNUMBER = {0217.16103},
      }
  • [JZ] Go to document W. B. Johnson and M. Zippin, "On subspaces of quotients of $(\sum G_{n})_{lp}$ and $(\sum G_{n})_{c_{0}}$," in Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, 1972, pp. 311-316 (1973).
    @inproceedings {JZ, MRKEY = {0331023},
      AUTHOR = {Johnson, W. B. and Zippin, M.},
      TITLE = {On subspaces of quotients of {$(\sum G_{n})_{lp}$} and {$(\sum G_{n})_{c_{0}}$}},
      BOOKTITLE = {Proceedings of the {I}nternational {S}ymposium on {P}artial {D}ifferential {E}quations and the {G}eometry of {N}ormed {L}inear {S}paces},
      VENUE={{J}erusalem, 1972},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {13},
      YEAR = {1972},
      PAGES = {311--316 (1973)},
      ISSN = {0021-2172},
      MRCLASS = {46B15},
      MRNUMBER = {0338745},
      MRREVIEWER = {J. A. Van Casteren},
      DOI = {10.1007/BF02762805},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0252.46025},
      }
  • [KP] M. I. Kadec and A. Pełczyński, "Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$," Studia Math., vol. 21, pp. 161-176, 1961/1962.
    @article {KP, MRKEY = {0152879},
      AUTHOR = {Kadec, M. I. and Pe{\l}czy{ń}ski, A.},
      TITLE = {Bases, lacunary sequences and complemented subspaces in the spaces {$L_{p}$}},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      VOLUME = {21},
      YEAR = {1961/1962},
      PAGES = {161--176},
      ISSN = {0039-3223},
      MRCLASS = {46.35 (46.10)},
      MRNUMBER = {27 \#2851},
      MRREVIEWER = {H. Umegaki},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0102.32202},
      }
  • [KW] Go to document N. J. Kalton and D. Werner, "Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces," J. Reine Angew. Math., vol. 461, pp. 137-178, 1995.
    @article {KW, MRKEY = {1324212},
      AUTHOR = {Kalton, Nigel J. and Werner, Dirk},
      TITLE = {Property {$(M)$},
      {$M$}-ideals, and almost isometric structure of {B}anach spaces},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {461},
      YEAR = {1995},
      PAGES = {137--178},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {46B20 (46B28 47D15)},
      MRNUMBER = {1324212},
      MRREVIEWER = {Gilles Godefroy},
      DOI = {10.1515/crll.1995.461.137},
      ZBLNUMBER = {0823.46018},
      }
  • [KOS] Go to document H. Knaust, E. Odell, and T. Schlumprecht, "On asymptotic structure, the Szlenk index and UKK properties in Banach spaces," Positivity, vol. 3, iss. 2, pp. 173-199, 1999.
    @article {KOS, MRKEY = {1702641},
      AUTHOR = {Knaust, H. and Odell, E. and Schlumprecht, Th.},
      TITLE = {On asymptotic structure, the {S}zlenk index and {UKK} properties in {B}anach spaces},
      JOURNAL = {Positivity},
      FJOURNAL = {Positivity. An International Journal devoted to the Theory and Applications of Positivity in Analysis},
      VOLUME = {3},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {173--199},
      ISSN = {1385-1292},
      CODEN = {PSITFL},
      MRCLASS = {46B03 (46B10 46B20)},
      MRNUMBER = {1702641},
      MRREVIEWER = {Yves Raynaud},
      DOI = {10.1023/A:1009786603119},
      ZBLNUMBER = {0937.46006},
      }
  • [KK] Go to document A. Koldobsky and H. König, "Aspects of the isometric theory of Banach spaces," in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 899-939.
    @incollection {KK, MRKEY = {1863709},
      AUTHOR = {Koldobsky, Alexander and K{ö}nig, Hermann},
      TITLE = {Aspects of the isometric theory of {B}anach spaces},
      BOOKTITLE = {Handbook of the Geometry of {B}anach Spaces, {V}ol. {\rm I}},
      PAGES = {899--939},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2001},
      MRCLASS = {46B04 (46B07)},
      MRNUMBER = {1863709},
      MRREVIEWER = {Beata Randrianantoanina},
      DOI = {10.1016/S1874-5849(01)80023-1},
      ZBLNUMBER = {1005.46005},
      }
  • [KM] Go to document J. -L. Krivine and B. Maurey, "Espaces de Banach stables," Israel J. Math., vol. 39, iss. 4, pp. 273-295, 1981.
    @article {KM, MRKEY = {636897},
      AUTHOR = {Krivine, J.-L. and Maurey, B.},
      TITLE = {Espaces de {B}anach stables},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {39},
      YEAR = {1981},
      NUMBER = {4},
      PAGES = {273--295},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {46B20},
      MRNUMBER = {636897},
      MRREVIEWER = {D. J. H. Garling},
      DOI = {10.1007/BF02761674},
      ZBLNUMBER = {0504.46013},
      }
  • [LP] J. Lindenstrauss and A. Pełczyński, "Absolutely summing operators in $L_{p}$-spaces and their applications," Studia Math., vol. 29, pp. 275-326, 1968.
    @article {LP, MRKEY = {0231188},
      AUTHOR = {Lindenstrauss, J. and Pe{\l}czy{ń}ski, A.},
      TITLE = {Absolutely summing operators in {$L_{p}$}-spaces and their applications},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      VOLUME = {29},
      YEAR = {1968},
      PAGES = {275--326},
      ISSN = {0039-3223},
      MRCLASS = {46.35},
      MRNUMBER = {37 \#6743},
      MRREVIEWER = {A. Pietsch},
      ZBLNUMBER = {0183.40501},
      }
  • [LR] Go to document J. Lindenstrauss and H. P. Rosenthal, "The $\mathcal{L}_{p}$ spaces," Israel J. Math., vol. 7, pp. 325-349, 1969.
    @article {LR, MRKEY = {0270119},
      AUTHOR = {Lindenstrauss, J. and Rosenthal, H. P.},
      TITLE = {The {$\mathcal{L}_{p}$} spaces},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {7},
      YEAR = {1969},
      PAGES = {325--349},
      ISSN = {0021-2172},
      MRCLASS = {46.10},
      MRNUMBER = {42 \#5012},
      MRREVIEWER = {A. Pelczy{ń}ski},
      DOI = {10.1007/BF02788865},
      ZBLNUMBER = {0205.12602},
      }
  • [Ma] Go to document D. A. Martin, "Borel determinacy," Ann. of Math., vol. 102, iss. 2, pp. 363-371, 1975.
    @article {Ma, MRKEY = {0403976},
      AUTHOR = {Martin, Donald A.},
      TITLE = {Borel determinacy},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {102},
      YEAR = {1975},
      NUMBER = {2},
      PAGES = {363--371},
      ISSN = {0003-486X},
      MRCLASS = {02K30 (90D05 04A15)},
      MRNUMBER = {53 \#7785},
      MRREVIEWER = {John P. Burgess},
      DOI = {10.2307/1971035},
      ZBLNUMBER = {0336.02049},
      }
  • [MMT] B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann, "Asymptotic infinite-dimensional theory of Banach spaces," in Geometric Aspects of Functional Analysis, Basel: Birkhäuser, 1995, vol. 77, pp. 149-175.
    @incollection {MMT, MRKEY = {1353458},
      AUTHOR = {Maurey, B. and Milman, V. D. and Tomczak-Jaegermann, N.},
      TITLE = {Asymptotic infinite-dimensional theory of {B}anach spaces},
      BOOKTITLE = {Geometric Aspects of Functional Analysis},
      VENUE={{I}srael, 1992--1994},
      SERIES = {Oper. Theory Adv. Appl.},
      VOLUME= {77},
      PAGES = {149--175},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1995},
      MRCLASS = {46B20},
      MRNUMBER = {1353458},
      MRREVIEWER = {M. I. Kadets},
      ZBLNUMBER = {0872.46013},
      }
  • [O] Go to document E. Odell, "On complemented subspaces of $(\sum l_{2})_{lp}$," Israel J. Math., vol. 23, iss. 3-4, pp. 353-367, 1976.
    @article {O, MRKEY = {0423050},
      AUTHOR = {Odell, E.},
      TITLE = {On complemented subspaces of {$(\sum l_{2})_{lp}$}},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {23},
      YEAR = {1976},
      NUMBER = {3-4},
      PAGES = {353--367},
      ISSN = {0021-2172},
      MRCLASS = {46B05},
      MRNUMBER = {54 \#11034},
      MRREVIEWER = {Robert E. Huff},
      DOI = {10.1007/BF02761814},
      ZBLNUMBER = {0333.46005},
      }
  • [OS1] Go to document E. Odell and T. Schlumprecht, "Trees and branches in Banach spaces," Trans. Amer. Math. Soc., vol. 354, iss. 10, pp. 4085-4108, 2002.
    @article {OS1, MRKEY = {1926866},
      AUTHOR = {Odell, E. and Schlumprecht, Th.},
      TITLE = {Trees and branches in {B}anach spaces},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {354},
      YEAR = {2002},
      NUMBER = {10},
      PAGES = {4085--4108},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {46B03},
      MRNUMBER = {1926866},
      MRREVIEWER = {Gilles Godefroy},
      DOI = {10.1090/S0002-9947-02-02984-7},
      ZBLNUMBER = {1023.46014},
      }
  • [OS2] E. W. Odell and T. Schlumprecht, "Embedding into Banach spaces with finite dimensional decompositions," RACSAM Rev. R. Acad. Cienc. Exactas F\’\i s. Nat. Ser. A Mat., vol. 100, iss. 1-2, pp. 295-323, 2006.
    @article {OS2, MRKEY = {2267413},
      AUTHOR = {Odell, Edward W. and Schlumprecht, Thomas},
      TITLE = {Embedding into {B}anach spaces with finite dimensional decompositions},
      JOURNAL = {RACSAM Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Mat.},
      FJOURNAL = {RACSAM. Revista de la Real Academia de Ciencias Exactas, F\'\i sicas y Naturales. Serie A. Matemáticas},
      VOLUME = {100},
      YEAR = {2006},
      NUMBER = {1-2},
      PAGES = {295--323},
      ISSN = {1578-7303},
      MRCLASS = {46B03 (46B20)},
      MRNUMBER = {2267413},
      MRREVIEWER = {George Androulakis},
      ZBLNUMBER = {1118.46018},
      }
  • [OSZ] Go to document E. Odell, T. Schlumprecht, and A. Zsák, "On the structure of asymptotic $l_ p$ spaces," Q. J. Math., vol. 59, iss. 1, pp. 85-122, 2008.
    @article {OSZ, MRKEY = {2392502},
      AUTHOR = {Odell, E. and Schlumprecht, Th. and Zs{á}k, A.},
      TITLE = {On the structure of asymptotic {$l_ p$} spaces},
      JOURNAL = {Q. J. Math.},
      FJOURNAL = {The Quarterly Journal of Mathematics},
      VOLUME = {59},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {85--122},
      ISSN = {0033-5606},
      MRCLASS = {46B03 (46B15)},
      MRNUMBER = {2392502},
      MRREVIEWER = {Pandelis Dodos},
      DOI = {10.1093/qmath/ham026},
      ZBLNUMBER = {1156.46013},
      }
  • [P] A. Pełczyński, "Projections in certain Banach spaces," Studia Math., vol. 19, pp. 209-228, 1960.
    @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},
      }
  • [PR] A. Pełczyński and H. P. Rosenthal, "Localization techniques in $L^{p}$ spaces," Studia Math., vol. 52, pp. 263-289, 1974/75.
    @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},
      }
  • [Ro] Go to document C. Rosendal, "Infinite asymptotic games," Ann. Inst. Fourier $($Grenoble$)$, vol. 59, iss. 4, pp. 1359-1384, 2009.
    @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},
      }
  • [R] Go to document H. P. Rosenthal, "On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables," Israel J. Math., vol. 8, pp. 273-303, 1970.
    @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},
      }
  • [S1] Go to document G. Schechtman, "Examples of $\mathcal{L}_{p}$ spaces $(1<p\not=2<\infty )$," Israel J. Math., vol. 22, iss. 2, pp. 138-147, 1975.
    @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},
      }
  • [S2] G. Schechtman.
    @misc{S2,
      author={Schechtman, G.},
      NOTE = {personal communication},
      }
  • [Z] Go to document M. Zippin, "Banach spaces with separable duals," Trans. Amer. Math. Soc., vol. 310, iss. 1, pp. 371-379, 1988.
    @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},
      }

Authors

Richard Haydon

Brasenose College
Oxford OX1 4AJ
U.K.

Edward Odell

The University of Texas at Austin
Austin, TX 78712

Thomas Schlumprecht

Brasenose College
Oxford OX1 4AJ
U.K.

Current address:

Department of Mathematics
Texas A&M University
College Station, TX 77843