Two-norm spaces and decompositions of Banach spaces. II

Authors:
P. K. Subramanian and S. Rothman

Journal:
Trans. Amer. Math. Soc. **181** (1973), 313-327

MSC:
Primary 46B15

MathSciNet review:
0320719

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Banach space, a closed subspace of . One says is *-reflexive* if the canonical imbedding of onto is an isometry and -*pseudo reflexive* if it is a linear isomorphism onto. If has a basis and is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for to be -pseudo reflexive are due to I. Singer. To every -space with a decomposition we associate a canonical two-norm space and show that the properties of , in particular its -completion, may be exploited to give different proofs of Singer's results and, in particular, to extend them to -spaces with decompositions. This technique is then applied to a study of direct sum of -spaces with respect to a *BK* space. Necessary and sufficient conditions for such a space to be reflexive are obtained.

**[1]**Leon Alaoglu,*Weak topologies of normed linear spaces*, Ann. of Math. (2)**41**(1940), 252–267. MR**0001455****[2]**A. Alexiewicz and Z. Semadeni,*The two-norm spaces and their conjugate spaces*, Studia Math.**18**(1959), 275–293. MR**0115075****[3]**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**0117531****[4]**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.**[5]**W. Orlicz and V. Pták,*Some remarks on Saks spaces*, Studia Math.**16**(1957), 56–68. MR**0094686****[6]**S. Rothman,*Banach spaces via two-norm spaces*, Dissertation, University of Wisconsin, Milwaukee, Wis., 1970.**[7]**William H. Ruckle,*The infinite sum of closed subspaces of an 𝐹-space*, Duke Math. J.**31**(1964), 543–554. MR**0166589****[8]**B. L. Sanders,*Decompositions and reflexivity in Banach spaces*, Proc. Amer. Math. Soc.**16**(1965), 204–208. MR**0172092**, 10.1090/S0002-9939-1965-0172092-8**[9]**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**0152859****[10]**P. K. Subramanian,*Two-norm spaces and decompositions of Banach spaces. I*, Studia Math.**43**(1972), 179–194. MR**0315414**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46B15

Retrieve articles in all journals with MSC: 46B15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0320719-9

Keywords:
Schauder basis,
Schauder decomposition,
reflexive,
pseudo reflexive,
two-norm space,
spaces with mixed topology

Article copyright:
© Copyright 1973
American Mathematical Society