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As earlier, I have received an answer from this site that Bolzano Weierstrass' theorem is true for finite dimensional normed spaces, but not for infinite dimensional spaces. This, in particular => all finite dim. normed spaces are complete(in the sense that every Cauchy sequence converges(w.r.t. norm)). However, is it true that every normed vector space is complete?

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    We wouldn't need the words "Banach space" if this were true :)2012-05-13
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    There a lot of examples. http://math.stackexchange.com/questions/114070/how-to-prove-that-ck-omega-is-not-complete/114131#1141312012-05-13
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    Ignore my previous (non-)answer. Since I don't want to bamboozle you with function spaces I'm now trying to think of a vector space that is incomplete and is not a function space.2012-05-13
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    This might be an easier example: http://math.stackexchange.com/questions/143857/c0-1-is-not-a-banach-space-w-r-t-cdot-2/143861#1438612012-05-13
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    @MichaelGreinecker I don't insist...2012-05-13

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