Preservation of Closure in a Locally Convex Space. I

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 HI 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. Introduction. An interesting but not well-known result of Y. I. Petunin [2, pp. 1160-1162] is the following: Let £ be a Banach space and S its unit ball. Then E is reflexive if and only if 5 is closed in every Hausdorff locally convex topology on E that is weaker than the initial topology of E. The proof Petunin has given of his original theorem relies heavily on the concept of a characteristic of a subspace. However, we have found that the essence of Petunin's theorem does not depend upon the concept of a characteristic nor on boundedness nor on convexity. It will be seen that linearity is what makes Petunin's theorem work. The form which Petunin's theorem now assumes gives little insight. Let us view Petunin's theorem from a different perspective. Reflexivity of E is equivalent to having 5 weakly compact. Also, 5 is weakly compact if and only if S is a(E", £")closed in E". Therefore, Petunin's theorem can be restated as follows: The unit ball S is a(E", £")-closed in E" if and only if S is closed in every Hausdorff locally convex topology on E that is weaker than the initial topology of E. It is this perspective that will prove to be very fruitful, in particular, in generalizing Petunin's theorem to arbitrary locally convex spaces. We wish now to abstract the above situation. Let H be a topological vector space and G a subspace. Let A be a proper subset of G. We wish to investigate under what conditions A closed in G implies A closed in H. The reformulation of Petunin's theorem suggests that we look for a certain class of linear topologies on G Received by the editors December 27, 1976 and, in revised form, March 16, 1977. A MS (MOS) subject classifications (1970). Primary 46A99, 46-02; Secondary 46A05, 46A20, 46A25.