Dilation Mapping on Topological Vector Space is Continuous

Theorem
Let $K$ be a 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 $X \to X$ with $x \mapsto \lambda x$ is continuous.

This is precisely $c_\lambda$.