Two-norm spaces and decompositions of Banach spaces. II
HTML articles powered by AMS MathViewer
- by P. K. Subramanian and S. Rothman PDF
- Trans. Amer. Math. Soc. 181 (1973), 313-327 Request permission
Abstract:
Let $X$ be a Banach space, $Y$ a closed subspace of ${X^\ast }$. One says $X$ is $Y$-reflexive if the canonical imbedding of $X$ onto ${Y^\ast }$ 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 ${X_s}$ and show that the properties of ${X_s}$, in particular its $\gamma$-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.References
- Leon Alaoglu, Weak topologies of normed linear spaces, Ann. of Math. (2) 41 (1940), 252–267. MR 1455, DOI 10.2307/1968829
- A. Alexiewicz and Z. Semadeni, The two-norm spaces and their conjugate spaces, Studia Math. 18 (1959), 275–293. MR 115075, DOI 10.4064/sm-18-3-275-293
- A. Alexiewicz and Z. Semadeni, Some properties of two-norm spaces and a characterization of reflexivity of Banach spaces, Studia Math. 19 (1960), 115–132. MR 117531, DOI 10.4064/sm-19-2-115-132 G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1960; English transl., Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR 24 #A411; MR 40 #1750.
- W. Orlicz and V. Pták, Some remarks on Saks spaces, Studia Math. 16 (1957), 56–68. MR 94686, DOI 10.4064/sm-16-1-56-68 S. Rothman, Banach spaces via two-norm spaces, Dissertation, University of Wisconsin, Milwaukee, Wis., 1970.
- William H. Ruckle, The infinite sum of closed subspaces of an $F$-space, Duke Math. J. 31 (1964), 543–554. MR 166589
- B. L. Sanders, Decompositions and reflexivity in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 204–208. MR 172092, DOI 10.1090/S0002-9939-1965-0172092-8
- Ivan Singer, On Banach spaces reflexive with respect to a linear subspace of their conjugate space. III, Rev. Math. Pures Appl. 8 (1963), 139–150. MR 152859
- P. K. Subramanian, Two-norm spaces and decompositions of Banach spaces. I, Studia Math. 43 (1972), 179–194. MR 315414, DOI 10.4064/sm-43-3-179-194
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 181 (1973), 313-327
- MSC: Primary 46B15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0320719-9
- MathSciNet review: 0320719