Translation Mapping on Topological Vector Space is Homeomorphism

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 a homeomorphism.

Proof
From Translation Mapping on Topological Vector Space is Continuous, both $T_x$ and $T_{-x}$ are continuous.

It is therefore sufficient to establish that $T_{-x}$ is the inverse mapping of $T_x$.

For all $y \in X$, we have:

and:

So both $T_x \circ T_{-x}$ and $T_{-x} \circ T_x$ are the identity mapping for $X$.

So $T_{-x}$ is the inverse mapping of $T_x$, as required.