if every continuous seminorm is bounded on Every totally bounded set is bounded. If is a vector subspace of a TVS then a subset of is bounded in if and only if it is bounded in Metrizability A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an F-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable. More strongly: a topological vector space is said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin. Let be a non-discrete locally compact topological field, for example the real or complex numbers. A Hausdorff topological vector space over is locally compact if and only if it is finite-dimensional, that is, isomorphic to for some natural number Completeness and uniform structure The canonical uniformity on a TVS is the unique translation-invariant uniformity that induces the topology on Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. This allows one to about related notions such as completeness, uniform convergence, Cauchy nets, and uniform continuity. etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is Tychonoff. A subspace of a TVS is compact if and only if it is complete and totally bounded (for Hausdorff TVSs, a set being totally bounded is equivalent to it being precompact). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are relatively compact). With respect to this uniformity, a net (or sequence) is Cauchy if and only if for every neighborhood of there exists some index such that whenever and Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called sequentially complete; in general, it may not be complete (in the sense that all Cauchy filters converge). The vector space operation of addition is uniformly continuous and an open map. Scalar multiplication is Cauchy continuous but in general, it