Abstract:This paper is concerned with the lifting of the closures of sets. If H is a topological vector space, G a subspace and A closed in G for the induced topology, under what conditions on A in G is it true that the closure of A is preserved in H, i.e., A is closed in H? In this paper a fundamental lifting proposition is proved. ’Preservation of closure’ will prove to be a fruitful technique in obtaining some interesting results in the theory of locally convex spaces. Using this technique, we will first show when closure is equivalent to completeness. Then we will prove a generalization to locally convex spaces of the classical Heine-Borel Theorem for Euclidean n-space. Generalizing a result of Petunin, we will also give some necessary and sufficient conditions on semireflexivity. Finally, we will give a necessary and sufficient condition for the sum of two closed subspaces to be closed.
J. Horváth, Topological vector spaces and distributions. I, Addison-Wesley, Reading, Mass., 1966.
Y. Petunin, Criterion for reflexivity of a Banach space, Soviet Math. Dokl. 2 (1961), 1160-1162.
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 262 (1980), 391-397
- MSC: Primary 46A05; Secondary 46A25
- DOI: https://doi.org/10.1090/S0002-9947-1980-0586723-4
- MathSciNet review: 586723