Abstract
The ℒ p spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒ p space if and only ifX* is and ℒ q space (p −1+q −1=1). (ii) A complemented subspace of an ℒ p space is either an ℒ p or an ℒ2 space. (iii) The ℒ p spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ∞ (resp. ℒ1) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space.
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References
C. Bessaga and A. Pełczyński,On bases and unconditional convergence of series in Banach spaces, Studia Math.17 (1958), 151–164.
C. Bessaga and A. Pełczyňski,Spaces of continuous functions IV, Studia Math.19 (1960), 53–62.
M. M. Day,Normed linear spaces, New York, 1962.
N. Dunford and J. T. Schwartz,Linear operators I, New York, 1958.
A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc.16 (1955).
A. Grothendieck,Une caracterisation vectorielle métrique des espaces L 1, Canad. J. Math.7 (1955), 552–561.
R. C. James,Uniformly non-square Banach spaces, Ann. of Math.80 (1964), 542–550.
M. I. Kadec and A. Pełczyňski,Bases, lacunary sequences and complemented subspaces in the spaces L p, Studia Math.21 (1962), 161–176.
S. Kakutani,Some characterizations of Euclidean spaces, Japan J. Math.16 (1939), 93–97.
V. Klee,On certain intersection properties of convex sets, Canad. J. Math.3 (1951), 272–275.
G. Köthe,Hebbare lokalkonvexe Raüme, Math. An.165 (1966), 181–195.
J. Lindenstrauss,On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J.10 (1963), 241–252.
J. Lindenstrauss,Extension of compact operators, Mem. Amer. Math. Soc.48 (1964).
J. Lindenstrauss,On a certain subspace of l 1, Bull. Acad. Polon. Sci.12 (1964), 539–542.
J. Lindenstrauss,On the extension of operators with a finite-dimensional range, Illinois J. Math.8 (1964), 488–499.
J. Lindenstrauss and A. Pełczynski,Absolutely summing operators in ℒ p spaces and their applications, Studia. Math.29 (1968), 275–326.
J. Lindenstrauss and H. P. Rosenthal,Automorphisms in c 0,l 1,and m, Israel J. Math.7 (1969), 227–239.
J. Lindenstrauss and D. Wulbert,On the classification of Banach spaces whose duals are L 1 spaces, J. Functional Analysis4 (1969), 332–349.
J. Lindenstrauss and M. Zippin,Banach spaces with sufficiently many Boolean algebras of Projections, J. Math. Anal. Appl.25 (1969), 309–320.
A. A. Milutin,Isomorphism of spaces of continuous functions on compacta of power continuum, Tieoria Funct., Funct. Anal. i Pril (Kharkov)2 (1966), 150–156 (Russian).
A. Pełczyňski,Projections in certain Banach spaces, Studia Math.19 (1960), 209–228.
A. Pełczyňski,Linear extensions, linear averagings and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math., n. 58 (1968).
H. P. Rosenthal,Projections onto translation-invariant subspaces of L p(G), Mem. Amer. Math. Soc.63 (1966).
H. P. Rosenthal, On injective Banach spaces and the spaces ℒ∞ (μ) for finite measures μ, (to appear).
W. Rudin,Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–227.
A. Sobczyk,Projections in Minkowski and Banach spaces, Duke Math. J.8 (1941), 78–106.
L. Tzafriri,An isomorphic characterization of L p and c 0 spaces, Studia Math.32 (1969), 286–295.
L. Tzafriri,Remarks on contractive projections in L p spaces, Israel J. Math.7 (1969), 9–15.
M. Zippin,On some subspaces of Banach spaces whose duals are L 1 spaces, Proc. Amer. Math. Soc.23 (1969) 378–385.
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This research was partially supported by NSF Grant# 8964.
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Lindenstrauss, J., Rosenthal, H.P. The ℒ p spaces. Israel J. Math. 7, 325–349 (1969). https://doi.org/10.1007/BF02788865
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DOI: https://doi.org/10.1007/BF02788865