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Two-Norm Spaces and Decompositions of Banach Spaces. II
P. K. Subramanian and S. Rothman
Transactions of the American Mathematical Society
Vol. 181 (Jul., 1973), pp. 313-327
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/1996635
Page Count: 15
You can always find the topics here!Topics: Topological theorems, Topology, Banach space, Mathematical sequences, Mathematical theorems, Unit vectors, Perceptron convergence procedure, Counterexamples, Isomorphism
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Let X be a Banach space, Y a closed subspace of X*. One says X is Y-reflexive if the canonical imbedding of X onto Y* is an isometry and Y-pseudo reflexive if it is a linear isomorphism onto. If X has a basis and Y is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for X to be Y-pseudo reflexive are due to I. Singer. To every B-space X with a decomposition we associate a canonical two-norm space Xs and show that the properties of Xs, in particular its γ-completion, may be exploited to give different proofs of Singer's results and, in particular, to extend them to B-spaces with decompositions. This technique is then applied to a study of direct sum of B-spaces with respect to a BK space. Necessary and sufficient conditions for such a space to be reflexive are obtained.
Transactions of the American Mathematical Society © 1973 American Mathematical Society