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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Reflexivity of $ L(E,\,F)$

Author: J. R. Holub
Journal: Proc. Amer. Math. Soc. 39 (1973), 175-177
MSC: Primary 46B10; Secondary 47D15
MathSciNet review: 0315407
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Abstract: Let $ E$ and $ F$ be Banach spaces and denote by $ L(E,F)$ (resp., $ K(E,F))$ the space of all bounded linear operators (resp., all compact operators) from $ E$ to $ F$. In this note the following theorem is proved: If $ E$ and $ F$ are reflexive and one of $ E$ and $ F$ has the approximation property then the following are equivalent:

(i) $ L(E,F)$ is reflexive,

(ii) $ L(E,F) = K(E,F)$,

(iii) if $ T \ne 0 \in L(E,F)$, then $ \vert\vert T\vert\vert = \vert\vert Tx\vert\vert$ for some $ x \in E,\vert\vert x\vert\vert = 1$.

This result extends a recent result of Ruckle (Proc. Amer. Math. Soc. 34 (1972), 171-174) who showed (i) and (ii) are equivalent when both $ E$ and $ F$ have the approximation property. Moreover the proof suggests strongly that the assumption of the approximation property may be dropped.

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Keywords: Spaces of linear operators, spaces of compact operators, norming point of an operator, reflexive space
Article copyright: © Copyright 1973 American Mathematical Society