Translation Mapping on Topological Vector Space is Continuous

Theorem

Let $K$ be a topological field.

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

Let $x \in X$.

Let $T_x$ be the translation by $x$ mapping.

Then $T_x$ is continuous.

Proof

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

From Horizontal Section of Continuous Function is Continuous, it follows that the $\paren {-x}$-horizontal section $T_x : X \to X$ with $y \mapsto y - x$ is continuous.

$\blacksquare$