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A closedness theorem for normed spaces

Abstract 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 Xd with respect to the space Y. If σ is a topology for Y, (compatible with the algebraic operations of Y), then the pointwise topology τp for YX is defined. We show that Xd is (algebraically) τp-closed in YX. For normed spaces is shown that suitable subspaces of Xd are τp-closed in a product space K ⊆ YX. As a corollary we obtain a generalization of Alaoglu’s theorem.