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 often the real or 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 trivial topology or indiscrete … 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, … See more