Definition:Locally Convex Space

Definition
Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $\mathcal P$ be a set of seminorms on $X$.

We call the pair $\struct {X, \mathcal P}$ a locally convex space over $\GF$.

Also see

 * Characterization of Locally Convex Spaces
 * Hausdorff Locally Convex Space is Topological Vector Space shows that Hausdorff locally convex spaces with their standard topology are topological vector spaces, and so theorems applicable to topological vector spaces may be used.