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Compactness and finite dimension in asymmetric normed linear spaces. (English) Zbl 1101.46017
Summary: We describe the compact sets of any asymmetric normed linear space. After that, we focus our attention on finite-dimensional asymmetric normed linear spaces. In this case, we establish the equivalence between the T 1 separation axiom and normable spaces. An asymmetric version of the Riesz theorem about the compactness of the unit ball is proved. We also prove that the Heine-Borel theorem characterizes finite-dimensional asymmetric normed linear spaces that satisfy the T 2 separation axiom. Finally, we focus our attention on the T 0 separation axiom and results that are related to the dual p-complexity spaces.

MSC:
46B99Normed linear spaces and Banach spaces
54E50Complete metric spaces
54H99Connections of general topology with other structures