Dilation Mapping on Topological Vector Space is Continuous

Theorem

Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field.

Let $\struct {X, +_X, \circ_X, \tau_X}$ be a vector space over $K$.

Let $\lambda \in K$.

Let $c_\lambda$ be the dilation by $\lambda$ mapping.

Then $c_\lambda$ is continuous.

Proof

Let $f: K \times X \to X$ be the map defined by:

$\tuple {\lambda, x} \mapsto \lambda \circ_X x$

for any $x \in X$

From the definition of a topological vector space, the mapping $f$ is continuous.

We are given that $c_\lambda$ is the dilation by $\lambda$ mapping.

That is, $c_\lambda : X \to X$ is the map:

$\forall x \in X: \map {c_\lambda} x = \lambda \circ_X x$

So $c_\lambda$ is by definition a $\lambda$-vertical section of $f$ such that:

$\forall x \in X: \map {c_\lambda} x = \map f {\lambda, x}$

The result follows from Vertical Section of Continuous Function is Continuous.

$\blacksquare$