# Is a topological subspace the same as a linear subspace in $\mathbb{R}^3$?

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
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
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