Normed spaces
Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a … See more

A topological vector space (TVS) $${\displaystyle X}$$ is a vector space over a topological field $${\displaystyle \mathbb {K} }$$ (most … See more

A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as … See more

Finest and coarsest vector topology
Let $${\displaystyle X}$$ be a real or complex vector space.
Trivial topology
The See more

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 … See more

Every topological vector space has a continuous dual space—the set $${\displaystyle X'}$$ of all continuous linear functionals, … See more

For any $${\displaystyle S\subseteq X}$$ of a TVS $${\displaystyle X,}$$ the convex (resp. balanced, disked, closed convex, closed balanced, closed disked') hull of $${\displaystyle S}$$ is the smallest subset of $${\displaystyle X}$$ that has this property and contains See more