Hereditary approximation property

Abstract

If $X$ is a Banach space such that the isomorphism constant to $\ell_2^n$ from $n$-dimensional subspaces grows sufficiently slowly as $n\to \infty$, then $X$ has the approximation property. A consequence of this is that there is a Banach space $X$ with a symmetric basis but not isomorphic to $\ell_2$ so that all subspaces of $X$ have the approximation property. This answers a problem raised in 1980. An application of the main result is that there is a separable Banach space $X$ that is not isomorphic to a Hilbert space, yet every subspace of $X$ is isomorphic to a complemented subspace of $X$. This contrasts with the classical result of Lindenstrauss and Tzafriri that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space.

  • [cgj] Go to document P. G. Casazza, C. L. Garc’ia, and W. B. Johnson, "An example of an asymptotically Hilbertian space which fails the approximation property," Proc. Amer. Math. Soc., vol. 129, iss. 10, pp. 3017-3023, 2001.
    @article {cgj, MRKEY = {1840107},
      AUTHOR = {Casazza, P. G. and Garc{\'ı}a, C. L. and Johnson, W. B.},
      TITLE = {An example of an asymptotically {H}ilbertian space which fails the approximation property},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {129},
      YEAR = {2001},
      NUMBER = {10},
      PAGES = {3017--3023},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {46B20 (46B07)},
      MRNUMBER = {1840107},
      MRREVIEWER = {Deba Prasad Sinha},
      DOI = {10.1090/S0002-9939-01-06142-1},
      ZBLNUMBER = {0983.46022},
      }
  • [cn] Go to document P. G. Casazza and N. J. Nielsen, "A Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2," Studia Math., vol. 157, iss. 1, pp. 1-16, 2003.
    @article {cn, MRKEY = {1980113},
      AUTHOR = {Casazza, Peter G. and Nielsen, Niels J.},
      TITLE = {A {B}anach space with a symmetric basis which is of weak cotype 2 but not of cotype 2},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Studia Mathematica},
      VOLUME = {157},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {1--16},
      ISSN = {0039-3223},
      MRCLASS = {46B03 (46B07)},
      MRNUMBER = {1980113},
      MRREVIEWER = {G. Schechtman},
      DOI = {10.4064/sm157-1-1},
      ZBLNUMBER = {1032.46011},
      }
  • [djt] J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge: Cambridge Univ. Press, 1995, vol. 43.
    @book {djt, MRKEY = {1342297},
      AUTHOR = {Diestel, Joe and Jarchow, Hans and Tonge, Andrew},
      TITLE = {Absolutely Summing Operators},
      SERIES = {Camb. Stud. Adv. Math.},
      VOLUME = {43},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {xvi+474},
      ISBN = {0-521-43168-9},
      MRCLASS = {46-02 (46Bxx 46M05 47-02 47B10 47D50)},
      MRNUMBER = {1342297},
      MRREVIEWER = {Andreas Defant},
      ZBLNUMBER = {0855.47016},
      }
  • [fj] T. Figiel and W. B. Johnson, "A uniformly convex Banach space which contains no $l_{p}$," Compositio Math., vol. 29, pp. 179-190, 1974.
    @article {fj, MRKEY = {0355537},
      AUTHOR = {Figiel, T. and Johnson, W. B.},
      TITLE = {A uniformly convex {B}anach space which contains no {$l\sb{p}$}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {29},
      YEAR = {1974},
      PAGES = {179--190},
      ISSN = {0010-437X},
      MRCLASS = {46B05},
      MRNUMBER = {0355537},
      MRREVIEWER = {A. Pelczynski},
      ZBLNUMBER = {0301.46013},
      }
  • [g] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires," Mem. Amer. Math. Soc., vol. 1955, iss. 16, p. 140, 1955.
    @article {g, MRKEY = {0075539},
      AUTHOR = {Grothendieck, Alexandre},
      TITLE = {Produits tensoriels topologiques et espaces nucléaires},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {1955},
      YEAR = {1955},
      NUMBER = {16},
      PAGES = {140},
      ISSN = {0065-9266},
      MRCLASS = {46.1X},
      MRNUMBER = {0075539},
      MRREVIEWER = {J. Sebasti{ã}o e Silva},
      ZBLNUMBER = {0123.30301},
      }
  • [ja] Go to document R. C. James, "Uniformly non-square Banach spaces," Ann. of Math., vol. 80, pp. 542-550, 1964.
    @article {ja, MRKEY = {0173932},
      AUTHOR = {James, Robert C.},
      TITLE = {Uniformly non-square {B}anach spaces},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {80},
      YEAR = {1964},
      PAGES = {542--550},
      ISSN = {0003-486X},
      MRCLASS = {46.10},
      MRNUMBER = {0173932},
      MRREVIEWER = {M. M. Day},
      DOI = {10.2307/1970663},
      ZBLNUMBER = {0132.08902},
      }
  • [j2] Go to document W. B. Johnson, "Factoring compact operators," Israel J. Math., vol. 9, pp. 337-345, 1971.
    @article {j2, MRKEY = {0290133},
      AUTHOR = {Johnson, William B.},
      TITLE = {Factoring compact operators},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {9},
      YEAR = {1971},
      PAGES = {337--345},
      ISSN = {0021-2172},
      MRCLASS = {47.10 (46.00)},
      MRNUMBER = {0290133},
      MRREVIEWER = {F. J. Murray},
      DOI = {10.1007/BF02771684},
      ZBLNUMBER = {0236.47045},
      }
  • [j1] W. B. Johnson, "Banach spaces all of whose subspaces have the approximation property," in Special Topics of Applied Mathematics, Amsterdam: North-Holland, 1980, pp. 15-26.
    @incollection {j1, MRKEY = {0585146},
      AUTHOR = {Johnson, William B.},
      TITLE = {Banach spaces all of whose subspaces have the approximation property},
      BOOKTITLE = {Special Topics of Applied Mathematics},
      VENUE={{P}roc. {S}em., {G}es. {M}ath. {D}atenverarb., {B}onn, 1979},
      PAGES = {15--26},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1980},
      MRCLASS = {46B20},
      MRNUMBER = {0585146},
      MRREVIEWER = {Pier Luigi Papini},
      ZBLNUMBER = {0442.46011},
      }
  • [k] Go to document W. B. Johnson, H. König, B. Maurey, and J. R. Retherford, "Eigenvalues of $p$-summing and $l_{p}$-type operators in Banach spaces," J. Funct. Anal., vol. 32, iss. 3, pp. 353-380, 1979.
    @article {k, MRKEY = {0538861},
      AUTHOR = {Johnson, William B. and K{ö}nig, H. and Maurey, B. and Retherford, J. R.},
      TITLE = {Eigenvalues of {$p$}-summing and {$l\sb{p}$}-type operators in {B}anach spaces},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {32},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {353--380},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {47B10 (47A70 47G05)},
      MRNUMBER = {0538861},
      MRREVIEWER = {A. Pietsch},
      DOI = {10.1016/0022-1236(79)90046-6},
      ZBLNUMBER = {0408.47019},
      }
  • [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, William 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},
      }
  • [js] Go to document W. B. Johnson and A. Szankowski, "Complementably universal {B}anach spaces," Studia Math., vol. 58, iss. 1, pp. 91-97, 1976.
    @article {js, MRKEY = {0425582},
      AUTHOR = {Johnson, William B. and Szankowski, A.},
      TITLE = {Complementably universal {B}anach spaces},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      VOLUME = {58},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {91--97},
      ISSN = {0039-3223},
      MRCLASS = {46B05},
      MRNUMBER = {0425582},
      MRREVIEWER = {M. I. Kadec},
      ZBLNUMBER = {0341.46017},
      URL = {http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-smv58i1p5bwm?q=6d7b6ba3-fd29-4b9b-b111-359913df3d0b$1&qt=IN_PAGE,
     
  • [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.
    @inproceedings {jz, MRKEY = {0331023},
      AUTHOR = {Johnson, William B. and Zippin, M.},
      TITLE = {On subspaces of quotients of {$(\sum G\sb{n})\sb{lp}$} and {$(\sum G\sb{n})\sb{c\sb{0}}$}},
      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},
      SERIES = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {13},
      YEAR = {1972},
      PAGES = {311--316},
      ISSN = {0021-2172},
      MRCLASS = {46B15},
      MRNUMBER = {0331023},
      MRREVIEWER = {J. A. Van Casteren},
      DOI = {10.1007/BF02762805},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0252.46025},
      }
  • [ka] Go to document M. I. Kadec, "On complementably universal {B}anach spaces," Studia Math., vol. 40, pp. 85-89, 1971.
    @article {ka, MRKEY = {0313764},
      AUTHOR = {Kadec, M. I.},
      TITLE = {On complementably universal {B}anach spaces},
      JOURNAL = {Studia Math.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
      VOLUME = {40},
      YEAR = {1971},
      PAGES = {85--89},
      ISSN = {0039-3223},
      MRCLASS = {46B15},
      MRNUMBER = {0313764},
      MRREVIEWER = {R. C. James},
      ZBLNUMBER = {0218.46015},
      URL = {http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-smv40i1p11bwm?q=7496c07b-dbb3-4aa0-b14a-0ca3367a9395$1&qt=IN_PAGE,
     
  • [l] V. B. Lidskiui, "Non-selfadjoint operators with a trace," Dokl. Akad. Nauk SSSR, vol. 125, pp. 485-487, 1959.
    @article {l, MRKEY = {0105023},
      AUTHOR = {Lidski{\u\i},
      V. B.},
      TITLE = {Non-selfadjoint operators with a trace},
      JOURNAL = {Dokl. Akad. Nauk SSSR},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {125},
      YEAR = {1959},
      PAGES = {485--487},
      ISSN = {0002-3264},
      MRCLASS = {46.00},
      MRNUMBER = {0105023},
      MRREVIEWER = {G.-C. Rota},
      ZBLNUMBER = {0104.33801},
     }
  • [lt3] Go to document J. Lindenstrauss and L. Tzafriri, "On the complemented subspaces problem," Israel J. Math., vol. 9, pp. 263-269, 1971.
    @article {lt3, MRKEY = {0276734},
      AUTHOR = {Lindenstrauss, Joram and Tzafriri, Lior},
      TITLE = {On the complemented subspaces problem},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {9},
      YEAR = {1971},
      PAGES = {263--269},
      ISSN = {0021-2172},
      MRCLASS = {46.10},
      MRNUMBER = {0276734},
      MRREVIEWER = {W. J. Davis},
      DOI = {10.1007/BF02771592},
      ZBLNUMBER = {0211.16301},
      }
  • [lt] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, New York: Springer-Verlag, 1977, vol. 92.
    @book {lt, MRKEY = {0500056},
      AUTHOR = {Lindenstrauss, Joram and Tzafriri, Lior},
      TITLE = {Classical {B}anach Spaces {I},
      Sequence Spaces},
      SERIES={Ergeb. Math. Grenzgeb.},
      VOLUME={92},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xiii+188},
      ISBN = {3-540-08072-4},
      MRCLASS = {46-02 (46A45 46BXX)},
      MRNUMBER = {0500056},
      MRREVIEWER = {S. V. Kisljakov},
      ZBLNUMBER = {0362.46013},
      }
  • [lt2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, New York: Springer-Verlag, 1979, vol. 97.
    @book {lt2, MRKEY = {0540367},
      AUTHOR = {Lindenstrauss, Joram and Tzafriri, Lior},
      TITLE = {Classical {B}anach Spaces {II},
      Function Spaces},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {97},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1979},
      PAGES = {x+243},
      ISBN = {3-540-08888-1},
      MRCLASS = {46-02 (46B30 46E30 46M05)},
      MRNUMBER = {0540367},
      MRREVIEWER = {A. C. Zaanen},
      ZBLNUMBER = {0403.46022},
      }
  • [o] Go to document J. Lindenstrauss and L. Tzafriri, "The uniform approximation property in Orlicz spaces," Israel J. Math., vol. 23, iss. 2, pp. 142-155, 1976.
    @article {o, MRKEY = {0399806},
      AUTHOR = {Lindenstrauss, Joram and Tzafriri, Lior},
      TITLE = {The uniform approximation property in {O}rlicz spaces},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {23},
      YEAR = {1976},
      NUMBER = {2},
      PAGES = {142--155},
      ISSN = {0021-2172},
      MRCLASS = {46B05 (46A25 46E30)},
      MRNUMBER = {0399806},
      MRREVIEWER = {William B. Johnson},
      DOI = {10.1007/BF02756794},
      ZBLNUMBER = {0347.46025},
      }
  • [m] V. Mascioni, "On Banach spaces isomorphic to their duals," Houston J. Math., vol. 19, iss. 1, pp. 27-38, 1993.
    @article {m, MRKEY = {1218076},
      AUTHOR = {Mascioni, Vania},
      TITLE = {On {B}anach spaces isomorphic to their duals},
      JOURNAL = {Houston J. Math.},
      FJOURNAL = {Houston Journal of Mathematics},
      VOLUME = {19},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {27--38},
      ISSN = {0362-1588},
      CODEN = {HJMADZ},
      MRCLASS = {46B10 (46C05)},
      MRNUMBER = {1218076},
      MRREVIEWER = {Giovanni Emmanuele},
      ZBLNUMBER = {0785.46010},
      }
  • [mt] Go to document P. Mankiewicz and N. Tomczak-Jaegermann, "Schauder bases in quotients of subspaces of $l_2(X)$," Amer. J. Math., vol. 116, iss. 6, pp. 1341-1363, 1994.
    @article {mt, MRKEY = {1305868},
      AUTHOR = {Mankiewicz, Piotr and Tomczak-Jaegermann, Nicole},
      TITLE = {Schauder bases in quotients of subspaces of {$l\sb 2(X)$}},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {116},
      YEAR = {1994},
      NUMBER = {6},
      PAGES = {1341--1363},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {46B15 (46B20 46C15 46E40)},
      MRNUMBER = {1305868},
      MRREVIEWER = {Wolfgang Lusky},
      DOI = {10.2307/2375049},
      ZBLNUMBER = {0827.46010},
     }
  • [nt] Go to document N. Nielsen and N. Tomczak-Jaegermann, "Banach lattices with property (H) and weak Hilbert spaces," Illinois J. Math., vol. 36, iss. 3, pp. 345-371, 1992.
    @article {nt, MRKEY = {1161972},
      AUTHOR = {Nielsen, NJ and Tomczak-Jaegermann, N.},
      TITLE = {Banach lattices with property ({H}) and weak {H}ilbert spaces},
      JOURNAL = {Illinois J. Math.},
      FJOURNAL = {Illinois Journal of Mathematics},
      VOLUME = {36},
      YEAR = {1992},
      NUMBER = {3},
      PAGES = {345--371},
      ISSN = {0019-2082},
      CODEN = {IJMTAW},
      MRCLASS = {46B42 (46B20 46C99)},
      MRNUMBER = {1161972},
      MRREVIEWER = {Yehoram Gordon},
      URL = {http://projecteuclid.org/euclid.ijm/1255987415},
      ZBLNUMBER = {0787.46018},
      }
  • [p0] Go to document G. Pisier, "Martingales with values in uniformly convex spaces," Israel J. Math., vol. 20, iss. 3-4, pp. 326-350, 1975.
    @article {p0, MRKEY = {0394135},
      AUTHOR = {Pisier, Gilles},
      TITLE = {Martingales with values in uniformly convex spaces},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {20},
      YEAR = {1975},
      NUMBER = {3-4},
      PAGES = {326--350},
      ISSN = {0021-2172},
      MRCLASS = {46B10 (60G45)},
      MRNUMBER = {0394135},
      MRREVIEWER = {S. D. Chatterji},
      DOI = {10.1007/BF02760337},
      ZBLNUMBER = {0344.46030},
      }
  • [p1] Go to document G. Pisier, "Weak Hilbert spaces," Proc. London Math. Soc., vol. 56, iss. 3, pp. 547-579, 1988.
    @article {p1, MRKEY = {0931514},
      AUTHOR = {Pisier, Gilles},
      TITLE = {Weak {H}ilbert spaces},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {56},
      YEAR = {1988},
      NUMBER = {3},
      PAGES = {547--579},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {46B20 (46C99 47B37)},
      MRNUMBER = {0931514},
      MRREVIEWER = {G. Schechtman},
      DOI = {10.1112/plms/s3-56.3.547},
      ZBLNUMBER = {0666.46009},
      }
  • [p2] Go to document G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge: Cambridge Univ. Press, 1989, vol. 94.
    @book {p2, MRKEY = {1036275},
      AUTHOR = {Pisier, Gilles},
      TITLE = {The Volume of Convex Bodies and {B}anach Space Geometry},
      SERIES = {Cambridge Tracts in Math.},
      VOLUME = {94},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1989},
      PAGES = {xvi+250},
      ISBN = {0-521-36465-5; 0-521-66635-X},
      MRCLASS = {52A21 (46B20 52A07)},
      MRNUMBER = {1036275},
      MRREVIEWER = {Mikhail Ostrovski{\u\i}},
      DOI = {10.1017/CBO9780511662454},
      ZBLNUMBER = {0698.46008},
      }
  • [px] Go to document G. Pisier and Q. H. Xu, "Random series in the real interpolation spaces between the spaces $v_p$," in Geometrical Aspects of Functional Analysis (1985/86), New York: Springer-Verlag, 1987, vol. 1267, pp. 185-209.
    @incollection {px, MRKEY = {0907695},
      AUTHOR = {Pisier, Gilles and Xu, Quan Hua},
      TITLE = {Random series in the real interpolation spaces between the spaces {$v\sb p$}},
      BOOKTITLE = {Geometrical Aspects of Functional Analysis (1985/86)},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1267},
      PAGES = {185--209},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {46Bxx (46E99 46M35)},
      MRNUMBER = {0907695},
      MRREVIEWER = {Irina U. Asekritova},
      DOI = {10.1007/BFb0078146},
      ZBLNUMBER = {0634.46009},
      }
  • [sc] Go to document T. Schlumprecht, "An arbitrarily distortable Banach space," Israel J. Math., vol. 76, iss. 1-2, pp. 81-95, 1991.
    @article {sc, MRKEY = {1177333},
      AUTHOR = {Schlumprecht, Thomas},
      TITLE = {An arbitrarily distortable {B}anach space},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {76},
      YEAR = {1991},
      NUMBER = {1-2},
      PAGES = {81--95},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {46B20 (46B03)},
      MRNUMBER = {1177333},
      MRREVIEWER = {Bor-Luh Lin},
      DOI = {10.1007/BF02782845},
      ZBLNUMBER = {0796.46007},
      }
  • [t] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Harlow: Longman Scientific & Technical, 1989.
    @book {t, MRKEY = {0993774},
      AUTHOR = {Tomczak-Jaegermann, Nicole},
      TITLE = {Banach-{M}azur Distances and Finite-Dimensional Operator Ideals},
      SERIES = {Pitman Monogr. Surveys Pure Appl. Math.},
      NUMBER = {38},
      PUBLISHER = {Longman Scientific \& Technical},
      ADDRESS = {Harlow},
      YEAR = {1989},
      PAGES = {xii+395},
      ISBN = {0-582-01374-7},
      MRCLASS = {46B20 (47B10 47D30)},
      MRNUMBER = {0993774},
      MRREVIEWER = {Yehoram Gordon},
      ZBLNUMBER = {0721.46004},
      }
  • [z] M. Zippin, "Interpolation of operators of weak type between rearrangement invariant function spaces," J. Functional Analysis, vol. 7, pp. 267-284, 1971.
    @article {z, MRKEY = {0412793},
      AUTHOR = {Zippin, M.},
      TITLE = {Interpolation of operators of weak type between rearrangement invariant function spaces},
      JOURNAL = {J. Functional Analysis},
      VOLUME = {7},
      YEAR = {1971},
      PAGES = {267--284},
      MRCLASS = {46E30 (46M35 47B37)},
      MRNUMBER = {0412793},
      ZBLNUMBER = {0224.46038},
      }

Authors

W. B. Johnson

Department of Mathematics
Texas A&M University
College Station, TX 77843-3368

A. Szankowski

The Hebrew University of Jerusalem
Jerusalem 91904
Israel