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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?2012-07-02
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    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.)2012-07-02
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    I always knew that $V$ was a vector space and I discovered that it has a topology too.2012-07-02
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    You might want to add more of your thoughts to stave off downvotes.2012-07-02
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    Hi, welcome to Math.SE. Your question, as stated, falls rather borderline on [this FAQ guideline](http://math.stackexchange.com/faq#dontask). I would suggest that you follow rschwieb's advice and provide more context to focus the scope of the question.2012-07-03
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    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](http://en.wikipedia.org/wiki/Hilbert_space). There is some synergy between topology and vector space properties (as more generally is the case for [topological groups](http://en.wikipedia.org/wiki/Topological_group)). But you've not given us much in the way of a question to answer. -12012-07-05

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