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Weakly Compact Sets in Orlicz Spaces

Published online by Cambridge University Press:  20 November 2018

Tsuyoshi Andô
Tsuyoshi Andô*
Affiliation:
Research Institute of Applied Electricity Hokkaidô University, SapporoJapan
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The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.

The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 170 - 176
Copyright
Copyright © Canadian Mathematical Society 1962

References

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