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The Dual space of an asymmetric normed linear space
Abstract
Given an asymmetric normed linear space (X, q), we construct and
study its dual space (X*, q*). In particular, we show that (x*, q*) is a
biBanach semilinear space and prove that (X, q) can be identified as a subspace
of its bidual by an isometric isomorphism.
We also introduce and characterize the so-called weak* topology which is
generated in a natural way by the relation between (X, q) and its dual, and an
extension of the celebrated Alaoglu's theorem is obtained.
Some parts of our theory are presented in the more general setting of the
space LC(X, Y) of all linear continuous mappings from the asymmetric normed
linear space X to the asymmetric normed linear space Y. In particular, we show
that LC(X, Y) can be endowed with the structure of an asymmetric normed
semilinear space and prove that it is a biBanach space if Y is so.
Mathematics Subject Classification (2000): 46B10, 54E50, 54E15, 54H99.
Key words: Asymmetric normed linear space; semilinear space; continuous
linear mapping; dual space; bidual space; biBanach space; quasi-metric; weak*
topology; compactness.
Quaestiones Mathematicae 26(2003), 83-96.
study its dual space (X*, q*). In particular, we show that (x*, q*) is a
biBanach semilinear space and prove that (X, q) can be identified as a subspace
of its bidual by an isometric isomorphism.
We also introduce and characterize the so-called weak* topology which is
generated in a natural way by the relation between (X, q) and its dual, and an
extension of the celebrated Alaoglu's theorem is obtained.
Some parts of our theory are presented in the more general setting of the
space LC(X, Y) of all linear continuous mappings from the asymmetric normed
linear space X to the asymmetric normed linear space Y. In particular, we show
that LC(X, Y) can be endowed with the structure of an asymmetric normed
semilinear space and prove that it is a biBanach space if Y is so.
Mathematics Subject Classification (2000): 46B10, 54E50, 54E15, 54H99.
Key words: Asymmetric normed linear space; semilinear space; continuous
linear mapping; dual space; bidual space; biBanach space; quasi-metric; weak*
topology; compactness.
Quaestiones Mathematicae 26(2003), 83-96.
