TVS need not be normal. Let be a topological vector space. Given a subspace the quotient space with the usual quotient topology is a Hausdorff topological vector space if and only if is closed. This permits the following construction: given a topological vector space (that is probably not Hausdorff), form the quotient space where is the closure of is then a Hausdorff topological vector space that can be studied instead of Invariance of vector topologies One of the most used properties of vector topologies is that every vector topology is : for all the map defined by is a homeomorphism, but if then it is not linear and so not a TVS-isomorphism. Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if then the linear map defined by is a homeomorphism. Using produces the negation map defined by which is consequently a linear homeomorphism and thus a TVS-isomorphism. If and any subset then and moreover, if then is a neighborhood (resp. open neighborhood, closed neighborhood) of in if and only if the same is true of at the origin. Local notions A subset of a vector space is said to be absorbing (in ): if for every there exists a real such that for any scalar satisfying balanced or circled: if for every scalar convex: if for every real a disk or absolutely convex: if is convex and balanced. symmetric: if or equivalently, if Every neighborhood of the origin is an absorbing set and contains an open balanced neighborhood of so every topological vector space has a local base of absorbing and balanced sets. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin. Bounded subsets A subset of a topological vector space is bounded if for every neighborhood of the origin, then when is sufficiently large. The definition of boundedness can be weakened a bit; is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also, is bounded if and only if for every balanced neighborhood of the origin, there exists such that Moreover, when is locally convex, the boundedness can be characterized by seminorms: the subset is bounded if and only