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By use of the theory of characters and the Pontryagin-van Kampen theorem, it is shown that if E is a locally compact vector space over a discrete division ring K of characteristic zero and if dimKE <2m, where m is the cardinality of K, then E is metrizable. The problem of determining whether a locally compact vector space over a discrete division ring is metrizable arises in the study of finite-dimensional locally compact vector spaces, because we have a fairly concrete picture of those that are metrizable: If E is a finitedimensional, metrizable, indiscrete locally compact vector space over a discrete field K and if o is the smallest open subspace of E, then the topological additive group o admits the structure of finite-dimensional topological vector space over the locally compact field F, where F is either the real field R, the field Qq of q-adic numbers, or the field Zp((X)) of power series over the field Z, of integers modulo p, under a scalar multiplication satisfying a. (Xx) =X(a.x) for all xCE, XCK, aC F; moreover, if E is a topological algebra, then o is an ideal and a. (xy) = (a.x)y, a. (yx) = y(a.x) for all aE F, xEo, yeA; finally, K is algebraically isomorphic to a subfield of finite codegree of a finite extension of F [4, Theorems 3 and 5]. Here we shall consider the special case of this problem where the scalar field has characteristic zero. First, we need a lower bound on the dimension of nonzero compact vector spaces. Let K be a division ring, equipped with the discrete topology. We denote by K^ the (compact) character group of the discrete additive group K, made into a right topological vector space over K by defining u.X:x l -u(\x) for all uCK , XCK, xEK [3, Theorem 1 ]. THEOREM 1. If K is an infinite division ring of cardinality m, then dimKKA = 2. PROOF. Case 1. The characteristic of K is zero. Then for some cardinal number n the additive group of K is isomorphic to Q(S), the Received by the editors January 20, 1970. AMS 1968 subject classifications. Primary 4601. Key wotds and phrases. Topological vector space, topological algebra, locally compact, metrizability. Copyright ? 1971, American Mathematical Society

Metrizability of locally compact vector spaces