A topological vector space is a vector space with a topology for which vector addition and scalar multiplication are continuous.
Every finite-dimensional real vector space has a unique Hausdorff topology making it a topological vector space. Choosing a basis identifies it with ordinary Euclidean space; every linear map between such spaces is continuous.
On a finite-dimensional vector space over a complete valued field, any two norms induce the same topology. Choosing a basis reduces the claim to comparison with the maximum of the absolute values of the coordinates.
A locally convex space is a topological vector space whose topology is generated by a family of seminorms. Its continuous dual consists of the linear functionals bounded by a finite maximum of those seminorms.
For an absorbing convex set containing zero, its Minkowski functional is . If is open and convex, then is sublinear and .
If is a nonempty open convex subset of a real locally convex space and , a continuous linear functional strictly separates them: after choosing a sign, for every . Apply the Hahn-Banach theorem to the Minkowski functional of a translate of .
If finitely many open convex subsets of a locally convex space have empty intersection, some continuous linear map preserves that fact: .
A continuous linear map is a linear map that is continuous for the topologies on its domain and codomain. Between normed vector spaces, continuity is equivalent to boundedness.
An isomorphism of Banach spaces is a bijective bounded linear map whose inverse is bounded. The bounded inverse theorem makes boundedness of the inverse automatic.
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A **topological vector space** is a type of vector space that is equipped with a topology, which allows for the definition of concepts such as convergence, continuity, and compactness in a way that is compatible with the vector space operations (vector addition and scalar multiplication).