Journal of Convex Analysis 22 (2015), No. 1, 037--060
Copyright Heldermann Verlag 2015
Norms with Infinite Values
Gerald Beer
Dept. of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A.
gbeer@cslanet.calstatela.edu
We provide an overview of linear spaces equipped with norms that may take on the value infinity but that otherwise satisfy the properties of ordinary norms. Spaces equipped with such norms fail to be topological vector spaces unless the norm is finite valued. We study the continuous linear transformations between such spaces, extending the theory of conventional linear analysis in often unanticipated ways.
Keywords: Extended norm, infinite valued norm, projection operator, projection complement, bornology, operator norm, effective domain, uniform convergence on bounded subsets, extended supremum norm, extended Lipschitz norm, distance basis.
MSC: 46B20, 46B28; 46A17, 46E15, 26A16
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