to in if and only if for every real number converges to in This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form with ). Finite-dimensional spaces By F. Riesz's theorem, a Hausdorff topological vector space is finite-dimensional if and only if it is locally compact, which happens if and only if it has a compact neighborhood of the origin. Let denote or and endow with its usual Hausdorff normed Euclidean topology. Let be a vector space over of finite dimension and so that is vector space isomorphic to (explicitly, this means that there exists a linear isomorphism between the vector spaces and ). This finite-dimensional vector space always has a unique vector topology, which makes it TVS-isomorphic to where is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) finest vector topology on has a unique vector topology if and only if If then although does not have a unique vector topology, it does have a unique vector topology. If then has exactly one vector topology: the trivial topology, which in this case (and in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension If then has two vector topologies: the usual Euclidean topology and the (non-Hausdorff) trivial topology. Since the field is itself a -dimensional topological vector space over and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an absorbing set and has consequences that reverberate throughout functional analysis. If then has distinct vector topologies: Some of these topologies are now described: Every linear functional on which is vector space isomorphic to induces a seminorm defined by where Every seminorm induces a (pseudometrizable locally convex) vector topology on and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on that are induced by linear functionals with distinct kernel will induces distinct vector topologies on However, while there are infinitely many vector topologies on when there are, only vector topologies on For instance, if then the vector topologies on consist of the trivial topology, the Hausdorff Euclidean