Vector Addition is Continuous in Weak Topology
Theorem
Let $K$ be a topological field.
Let $X$ be a topological vector space over $K$ with weak topology $w$.
Define $s : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$ by:
- $\map s {x, y} = x + y$
for each $x, y \in X$.
Then $s$ is continuous.
That is, vector addition remains continuous when restricting to the weak topology.
Proof
Let $X^\ast$ be the topological dual space of $X$.
From Continuity in Initial Topology, it suffices to show that for each $f \in X^\ast$ we have:
- $f \circ s : \struct {X, w} \times \struct {X, w} \to K$ is continuous.
Define the projections $\pr_1 : \struct {X, w} \times {X, w} \to \struct {X, w}$ and $\pr_2 : \struct {X, w} \times {X, w} \to \struct {X, w}$ as the projection onto the first and second factors.
Then for each $x, y \in X$ we have:
\(\ds \map {\paren {f \circ s} } {x, y}\) | \(=\) | \(\ds \map f {x + y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f x + \map f y\) | Definition of Linear Functional | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f {\map {\pr_1} {\tuple {x, y} } } + \map f {\map {\pr_2} {\tuple {x, y} } }\) |
That is:
- $f \circ s = f \circ \pr_1 + f \circ \pr_2$
From the definition of the product topology:
- $\pr_1 : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$
and:
- $\pr_2 : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$
are continuous.
From the definition of the weak topology, $f : \struct {X, w} \to K$ is continuous.
From Composite of Continuous Mappings is Continuous, $f \circ \pr_1 : \struct {X, w} \times \struct {X, w} \to K$ and $f \circ \pr_2 : \struct {X, w} \times \struct {X, w} \to K$ are continuous.
From Sum of Continuous Functions on Topological Ring is Continuous, $f \circ \pr_1 + f \circ \pr_2 : \struct {X, w} \times \struct {X, w} \to K$ is continuous.
So we have that $f \circ s : \struct {X, w} \to K$ is continuous for each $f \in X^\ast$.
So from Continuity in Initial Topology, $s : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$ is continuous in the weak topology.
$\blacksquare$