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I am having a bit of trouble wrapping my head around the difference between linear and topological subspaces. For example, say we have the z-axis missing. Then $\mathbb{R}^3$ without it is not a linear subspace, but it seems to me that it is an open topological subspace of $\mathbb{R}^3$. However, I saw somewhere that in all Hilbert spaces the two are the same. Now, I don't know much about Hilbert spaces, but it seems $\mathbb{R}^3$ is an example of it, so my conclusion would have to be wrong.

So what am I missing?

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Every subset of a topological space is a topological space. Thus, every non-trivial topological vector space has topological subspaces that are not vector subspaces. This includes Hilbert spaces. –  Brian M. Scott Feb 3 '13 at 4:04
    
I see. So just to get this straight then - the example I gave, i.e. $\mathbb{R}^3$ sans the z-axis, is a topological subspace, right? –  Ryker Feb 3 '13 at 4:07
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Absolutely. Everything that you said is correct except the bit that you said you saw somewhere. –  Brian M. Scott Feb 3 '13 at 4:09
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I would add that we could remove "non-trivial" from Brian's comment. For every topological vector space, the empty set is a topological subspace that is not a vector subspace. –  Jonas Meyer Feb 4 '13 at 2:51
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"However, I saw somewhere that in all Hilbert spaces the two are the same." They are never the same, for any topological vector space. Linear subspaces must be (nonempty and) closed under the vector space operations. Topological subspaces can be arbitrary as subsets.

What you may have seen is a common convention of using only the word "subspace" to refer to a particular type of subspace, where it is to be understood from context (or stated somewhere earlier in the reference), what is meant by that word. In the context of Hilbert space, often "subspace" automatically refers to linear subspace, and in some references it automatically refers to closed linear subspace (which makes a difference in infinite dimensions).

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link Here's where I saw this, but I might have misinterpreted what was meant by "vector subspace". I assumed it was a linear subspace. –  Ryker Feb 4 '13 at 6:48
    
@Ryker: I don't agree with Dan's comment there. "Vector subspace" can generally be safely assumed to mean the same thing as "linear subspace". While what I wrote in the second paragraph of my answer doesn't address that confusion, perhaps it is still a good warning for future reading. –  Jonas Meyer Feb 4 '13 at 6:54
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