Convex linear metric spaces are normable

Abstract

A linear metric space (X, d) is called a convex linear metric space if for all x, y in X, it also satisfies $$d(\lambda x+(1-\lambda )y,0)\le \lambda d(x,0)+(1-\lambda )d(y,0)$$ whenever $$0\le \lambda \le 1$$ . Such spaces, known to be more general than normed linear spaces, are examples of convex metric spaces extensively discussed in the literature. In this article, we show that convex linear metric spaces are infact normable.