Lattices and duality over local fields

Abstract

In this § and the next one, K will be a p-field, commutative or not. We shall mostly discuss only left vector-spaces over K; everything will apply in an obvious way to right vector-spaces. Only vector-spaces of finite dimension will occur; it is understood that these are always provided with their “natural topology” according to corollary 1 of th. 3, Chap. I-2. By th. 3 of Chap. I–2, every subspace of such a space V is closed in V. Taking coordinates, one sees that all linear mappings of such spaces into one another are continuous ; in particular, linear forms are continuous. Similarly, every injective linear mapping of such a space V into another is an isomorphism of V onto its image. As K is not compact, no subspace of V can be compact, except {0}.