Problem Suppose $R$ is a normed linear space, then show that:
If $M$ is closed subspace of $R$ and $N$ a finite dimensional subspace of $R$, then the set
$$M+N=\{ z : z = x + y , x \in M , y \in N \}$$
is a closed subspace of $R$.
What I've done I know being finite dimensional makes $N$ closed. Also I can disjointize $M$ and $N$ so that I have a direct sum instead of sum. But I don't know if the direct sum of closed subspaces are closed. I try taking a convergent sequence but i cant control the limit point. I even tried induction but cant show the 1-dimensional case. Could you please help me?
Best Answer
Look at the quotient space $X/M$. Because $M$ is known to be closed, then $Y=X/M$ is a normed space under the quotient norm. Let $\phi : X\rightarrow X/M$ be the quotient map. Then $\phi$ is continuous because $\|\phi(x)\|_{X/M}\le \|x\|_{X}$. Furthermore, $\phi(M+N)$ is finite-dimensional and, hence, closed in $X/M$. Therefore $M+N=\phi^{-1}(\phi(M+N))$ is closed.