F-spaces

locally convex space that is complete with respect to a translation-invariant metric

In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex. Addition in X is continuous with respect to d. The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. The metric space (X, d) is complete.