(TVS) is a vector space over a topological field (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition and scalar multiplication are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a or a on Every topological vector space is also a commutative topological group under addition. Hausdorff assumption Many authors (for example, Walter Rudin), but not this page, require the topology on to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below. Category and morphisms The category of topological vector spaces over a given topological field is commonly denoted or The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another. A (abbreviated ), also called a , is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when which is the range or image of is given the subspace topology induced by A (abbreviated ), also called a , is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. A (abbreviated ), also called a or an , is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection of subsets of a vector space is called if for every there exists some such that All of the above conditions are consequently a necessity for a topology to form a vector topology. Defining topologies using neighborhoods of the origin Since every vector topology is translation invariant (which means that for all the map defined by is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin. In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a