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?
Completeness of normed spaces
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linear-algebra
real-analysis
general-topology
functional-analysis
metric-spaces
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3We wouldn't need the words "Banach space" if this were true :) – 2012-05-13
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1There a lot of examples. http://math.stackexchange.com/questions/114070/how-to-prove-that-ck-omega-is-not-complete/114131#114131 – 2012-05-13
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0Ignore 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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0This might be an easier example: http://math.stackexchange.com/questions/143857/c0-1-is-not-a-banach-space-w-r-t-cdot-2/143861#143861 – 2012-05-13
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0@MichaelGreinecker I don't insist... – 2012-05-13