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I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?

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Over what field? – hardmath Jul 2 '12 at 11:54
Did you discover a vector space $V$ had a topology or did you discover a topological space $V$ was a vector space? (I'm assuming it was the former, but I'm just curious to see if my guess is right.) – rschwieb Jul 2 '12 at 12:18
You might want to add more of your thoughts to stave off downvotes. – rschwieb Jul 2 '12 at 14:19
Hi, welcome to Math.SE. Your question, as stated, falls rather borderline on this FAQ guideline. I would suggest that you follow rschwieb's advice and provide more context to focus the scope of the question. – Willie Wong Jul 3 '12 at 9:21
If $V$ is a vector space over the reals $\mathbb{R}$, then your "discovery" it has a topology is not much to write (or ask) about. The finite-dimensional real vector spaces are also known as Euclidean spaces, and these are the simplest examples of Hilbert spaces. There is some synergy between topology and vector space properties (as more generally is the case for topological groups). But you've not given us much in the way of a question to answer. -1 – hardmath Jul 5 '12 at 3:07

By having both the vector space and topological structures interacting, you will have the entire theory of topological vector spaces at hand. This is more than just the theory of vector spaces and theory of topological spaces acting independently.

One thing, for example, is that they are uniform spaces. This means essentially that the neighborhoods of any two points look the same.

Another thing is that even if you only require the separation axiom $T_0$, because of the way the vector space interacts with the topology, the topology must actually even be $T_{3\frac{1}{2}}$! If it's a finite dimensional vector space, you also have that $V$ is locally compact.

Another thing is that the continuous space of linear functionals $V^\ast$ gives rise to another interesting topology on $V$.

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