1. ## Closed Direct Sum

Hello all, I am having trouble with this problem and was wondering if anyone can help. I will post my partial proof and put the questions in bold, thanks very much!

Let H be a Hilbert space and let V,W be linear subspaces such that $V \cap W = \{0\}$. Prove that $V\oplus W$ is closed $\iff V,W$ are both closed.

Proof
( $\impliedby$) Let $V,W$ both be closed. Let $x_n$ be a convergent sequence in $V \oplus W$ with $x_n \rightarrow L \in H$. We have $x_n = v_n + w_n$ where $v_n$ is a sequence in $V$ and $w_n$ is a sequence in $W$. Now if $v_n$ and $w_n$ both converge then $v_n \rightarrow L_v$ and $w_n \rightarrow L_w$, where $L_v + L_w = L$, but what if $v_n$ and $w_n$ are divergent? Is it not possible they diverge but both somehow converge to $L$ when you add them together?

( $\implies$) Let $V \oplus W$ be closed. Let $v_n$ be a sequence in $V \oplus W$. Then $v_n = v_n + 0 \in V \oplus W$, so $v_n \rightarrow L \in V \oplus W$. How do I show that $L \in V$ ?

I would really appreciate any help with this, thank you

2. Originally Posted by slevvio
Hello all, I am having trouble with this problem and was wondering if anyone can help. I will post my partial proof and put the questions in bold, thanks very much!

Let H be a Hilbert space and let V,W be linear subspaces such that $V \cap W = \{0\}$. Prove that $V\oplus W$ is closed $\iff V,W$ are both closed.

Proof
( $\impliedby$) Let $V,W$ both be closed. Let $x_n$ be a convergent sequence in $V \oplus W$ with $x_n \rightarrow L \in H$. We have $x_n = v_n + w_n$ where $v_n$ is a sequence in $V$ and $w_n$ is a sequence in $W$. Now if $v_n$ and $w_n$ both converge then $v_n \rightarrow L_v$ and $w_n \rightarrow L_w$, where $L_v + L_w = L$, but what if $v_n$ and $w_n$ are divergent? Is it not possible they diverge but both somehow converge to $L$ when you add them together?

( $\implies$) Let $V \oplus W$ be closed. Let $v_n$ be a sequence in $V \oplus W$. Then $v_n = v_n + 0 \in V \oplus W$, so $v_n \rightarrow L \in V \oplus W$. How do I show that $L \in V$ ?

I would really appreciate any help with this, thank you
Are you aware of the result that under these circumstances $\displaystyle \pi_{V}$ and $\pi_{W}$ are continuous? This is a result of the closed graph theorem.

3. Ah thanks I was not, but I cannot use that formula to solve this if it was on the exam! i will investigate that theorem though