Locally convex space

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 $U$ containing zero, its Minkowski functional is $p_U(x)=\inf \{t>0:x\in tU\}$. If $U$ is open and convex, then $p_U$ is sublinear and $U=\{x:p_U(x)<1\}$.

If $C$ is a nonempty open convex subset of a real locally convex space and $x_0\notin C$, a continuous linear functional strictly separates them: after choosing a sign, $f(c)<f(x_0)$ for every $c\in C$. Apply the Hahn-Banach theorem to the Minkowski functional of a translate of $C$.

If finitely many open convex subsets $K_1,\dots ,K_n$ of a locally convex space have empty intersection, some continuous linear map $T:X\to {\mathbb{R}}^{n-1}$ preserves that fact: ${\bigcap }_{i}T(K_i)=\varnothing$.

The product topology on $X\times Y$ is generated by the seminorms $(x,y)\mapsto p(x)$ and $(x,y)\mapsto q(y)$. Its continuous dual is canonically $X^{\ast }\oplus Y^{\ast }$.