Prove: sum of a finite dim. subspace with a subspace is closed

[CornMuffin]

Homework Statement

Prove:
If $X$ is a (possibly infinite dimensional) locally convex space, $L \leq X$, $dimL < \infty$, and $M \leq X$ then $L + M$ is closed.

Homework Equations

The Attempt at a Solution

$dimL < \infty \implies L$ is closed in $X$
$L+M = \{ x+y : x\in L, y \in M \} \implies ^{??} dim(L+M) < \infty \implies L+M$ is closed in $X$

Homework Statement

Homework Equations

The Attempt at a Solution

 

[micromass]

Don't you need M to be closed as well??

Anyway, your attempt isn't correct since L+M doesn't need to be finite-dimensional.