# Dilation Mapping on Topological Vector Space is Continuous

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## Theorem

Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $\lambda \in K$.

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

Then $c_\lambda$ is continuous.

## Proof

From the definition of a topological vector space, the mapping $K \times X \to X$ defined by $\tuple {\lambda, x} \mapsto \lambda x$ is continuous.

From Vertical Section of Continuous Function is Continuous, it follows that the $\lambda$-vertical section $c_\lambda : X \to X$ with $x \mapsto \lambda x$ is continuous.

$\blacksquare$