Dilation of Compact Set in Topological Vector Space is Compact/Proof 2

Theorem

Let $k$ be a topological field.

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

Let $K$ be a compact subset of $X$.

Let $t \in k \setminus \set {0_k}$.

Then $t K$ is compact.

Proof

From Dilation Mapping on Topological Vector Space is Continuous, the mapping $c_t : X \to X$ defined by:

$\map {c_t} x = t x$

for each $x \in X$ is continuous.

From Continuous Image of Compact Space is Compact:

$\map {c_t} K = t K$ is compact.

$\blacksquare$