What is a Closed Linear Subspace?

[mynameiseva]

Hi. I'm trying to find a good definition of a closed linear subspace (as opposed to any other linear subspace), and I can't find anything concise and comprehensible. Any help will be much appreciated.
P.S. I'm not great at analysis, so please try to keep it simple.

 

[HallsofIvy]

"closed" in what sense? Closure under vector addition and scalar multiplication are part of the definition of "subspace". Topological closure depends upon having a topology on the space.

 

[mynameiseva]

Yeah, that's what I mean. I don't understand why people talk about 'closed linear subspaces' when every linear subspace is closed under scalar multiplication and vector addition. Here's an example,
"If L is a closed linear subspace of H, then the set of of all vectors in H that are orthogonal to every vector in L is itself a closed linear subspace".
But 'closed linear subspace' definitely means something different to just 'linear subspace', because the authors only describe some linear subspaces as 'closed'.

 

[micromass]

Hi mynameiseva!

Yeah, that's what I mean. I don't understand why people talk about 'closed linear subspaces' when every linear subspace is closed under scalar multiplication and vector addition. Here's an example,
"If L is a closed linear subspace of H, then the set of of all vectors in H that are orthogonal to every vector in L is itself a closed linear subspace".
But 'closed linear subspace' definitely means something different to just 'linear subspace', because the authors only describe some linear subspaces as 'closed'.

Judging from your quote, you are working in a Hilbert space H. A set L in a Hilbert space is called closed if

For all sequences $(x_n)_n$ in L such that $x_n\rightarrow x$ in H, then x is in L.​

Thus all sequences in L that converge, will converge to points in L. My professor once made the comparison to a prison: "a closed set is like a prison, you can't get out of it, not even in the limit".

A closed subspace is now simply a subspace that is closed. Note that closed subspaces are Hilbert spaces in their own right!

 

[mynameiseva]

Thanks a lot. I think I understand now.