Sum of closed linear subspaces necessarily closed?

Asked Sep 26 '14 at 07:52

Active Sep 26 '14 at 07:52

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Let $H$ be an infinite-dimensional Hilbert space. Let $L_1,L_2 \subset H$ be two closed linear subspaces.

If it is also known that $L_1 \perp L_2$ then it is not hard to show that $L_1 + L_2 = \{x_1 + x_2 | x_1 \in L_1, x_2 \in L_2 \}$ is also closed.

Is that also this way when they are not necessarily orthogonal? I think that the answer is "no", but can not find a good counter-example.

asked Sep 26 '14 at 07:52

Aahz

If $y_n = (x_1)_n+a(x_2)_n$ then the limit as n goes to infinity of $y_n$ is $X_1+aX_2$ where $X_1$ and $X_2$ are in $L_1$ and $L_2$ respectively, as those subspaces are closed. So $L_1 + L_2$ is closed, no? – Paul Sep 26 '14 at 10:44
But how do you know what the limit of $y_n$ is? – Aahz Sep 26 '14 at 11:08
http://math.stackexchange.com/questions/135471/the-direct-sum-of-two-closed-subspace-is-closed-hilbert-space – daw Sep 27 '14 at 15:51
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Possible duplicate of The direct sum of two closed subspace is closed? (Hilbert space) – Hanno Feb 17 '19 at 22:07

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