Topological Vector Spaces/Elementary constructions

Proposition (Topological vector space with the initial topology):

Let $E$ be a vector space, let $(E_{i})_{i\in I}$ be a family of topological vector spaces and let $f_{i}:E\to E_{i}$ be a family of linear functions. Then $E$ together with the initial topology induced by the $f_{i}$ is a topological vector space.

Proof: We are to show that scalar multiplication $\cdot :\mathbb {K} \times E\to E$ and addition $+:E\times E\to E$ are continuous. But this follows immediately, since a function $f:F\to E$ from a topological space $F$ to $E$ is continuous under the initial topology if and only if for all $i\in I$ the function $f_{i}\circ f$ is continuous (see General topology/ $\Box$

Dual systems, the weak topology

Definition (duality of vector spaces):

Let $E,E'$ be vector spaces over the field $\mathbb {K}$. $E,E'$ are set in duality iff there exists a bilinear function

$\langle \cdot ,\cdot \rangle :E\times E'\to \mathbb {K}$.