A CLOSEDNESS THEOREM FOR NORMED SPACES

TL;DRAbstract

For spaces X, Y, for which some algebraic operations are defined and in some cases topologies for X, Y are defined too, we define for the space X a dual space X d with respect to the space Y.If a is a topology for Y, (compatible with the algebraic operations of Y), then the pointwise topology r p for Y x is defined.We show that X d is (algebraically)r p -closed in Y x .For normed spaces is shown that suitable subspaces of X d are r p -closed in a product space K C Y x .As a corollary we obtain a generalization of Alaoglu's theorem.

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