1
$\begingroup$

Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.

What I know: If $L$ is a finite dimensional subspace, then $L$ is closed.

  • 0
    **Hint:** $L+M = \pi^{-1}[\pi(L)]$ if $\pi\colon X \to X/M$ is the quotient map.2012-06-05

1 Answers 1

4

If $M$ is not closed then this is not true in general. Indeed, consider $ X=\ell_1(\mathbb{N}),\quad L=\mathrm{span}\{e_1\},\quad M=\mathrm{span}\{e_n:n>1\} $ Obviously, $ \mathrm{dim}(L)=1,\quad \overline{M}=\{x\in X: x(1)=0\} $ $ L+M=\mathrm{span}\{e_n:n\in\mathbb{N}\} $ $ \overline{L+M}=\ell_1(\mathbb{N}) $

If we additionally assume that $M$ is closed then this statement is true. This proof is taken from Rudin's Functional analysis theorem 1.42. Since $M$ is closed then $X/M$ is Hausdorff locally convex space. Consider quotient map $ \pi:X\mapsto X/M:x\mapsto x+M $ Since $L$ is finite dimensional, then does $\pi(L)$. Since $\pi(L)$ is finite dimensional then it is closed. Since $\pi$ is continuous then $\pi^{-1}(\pi(L))$ is closed. It is remains to note that $M+L=\pi^{-1}(\pi(L))$