Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a See more

Throughout, $${\displaystyle {\mathcal {P}}}$$ is a family of continuous seminorms that generate the topology of $${\displaystyle X.}$$ See more

Finest and coarsest locally convex topology
Coarsest vector topology
Any vector space $${\displaystyle X}$$ endowed with the See more

• Convex metric space – metric space with the property any segment joining two points in that space has other points in it besides the endpoints
• Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points See more

Suppose $${\displaystyle X}$$ is a vector space over $${\displaystyle \mathbb {K} ,}$$ a subfield of the complex numbers (normally See more

• A family of seminorms $${\displaystyle \left(p_{\alpha }\right)_{\alpha }}$$ is called total or separated or is said to separate points if whenever $${\displaystyle p_{\alpha }(x)=0}$$ holds … See more

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally … See more