Sum of Compact Subsets of Topological Vector Space is Compact

Theorem

Let $K$ be a topological field.

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

Let $A$ and $B$ be compact (topological) subspaces of $X$.

Then $A + B$ is compact.

Proof

Since $X$ is a topological vector space, the map $+ : X \times X \to X$ defined by:

$\map + {x, y} = x + y$

for each $x, y \in X$ is continuous.

We have:

$A + B = + \sqbrk {A \times B}$

From Tychonoff's Theorem, $A \times B$ is compact.

From Continuous Image of Compact Space is Compact, $+ \sqbrk {A \times B} = A + B$ is therefore compact.

$\blacksquare$