topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on are all TVS-isomorphic to one another. Non-vector topologies Discrete and cofinite topologies If is a non-trivial vector space (that is, of non-zero dimension) then the discrete topology on (which is always metrizable) is a TVS topology because despite making addition and negation continuous (which makes it into a topological group under addition), it fails to make scalar multiplication continuous. The cofinite topology on (where a subset is open if and only if its complement is finite) is also a TVS topology on Linear maps A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator is continuous if is bounded (as defined below) for some neighborhood of the origin. A hyperplane on a topological vector space is either dense or closed. A linear functional on a topological vector space has either dense or closed kernel. Moreover, is continuous if and only if its kernel is closed. Types Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the open mapping theorem, and the fact that the dual space of the space separates points in the space. Below are some common topological vector spaces, roughly in order of increasing "niceness." F-spaces are complete topological vector spaces with a translation-invariant metric. These include spaces for all Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms. Local convexity is the minimum requirement for "geometrical" arguments like the Hahn–Banach theorem. The spaces are locally convex (in fact, Banach spaces) for all but not for Barrelled spaces: locally convex spaces where the Banach–Steinhaus theorem holds. Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators. Stereotype space: a locally convex space satisfying a variant of reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets. Montel space: a barrelled space where every closed and bounded