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.
$\blacksquare$