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    Let (P([0,1]),||\cdot ||_{\infty}) be the space of polynomials on [0,1] with the supremum norm. Show that (P([0,1]),||\cdot ||_{\infty}) is not complete
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    (Original post by gemma331)
    Let (P([0,1]),||\cdot ||_{\infty}) be the space of polynomials on [0,1] with the supremum norm. Show that (P([0,1]),||\cdot ||_{\infty}) is not complete
    Work backwards. Instead of trying to think of a cauchy sequence of polynomials that doesn't have a limit in your space, think about approximating other functions with polynomials... for instance you should have a result (else look it up) that tells you that P([0,1]) is dense in C([0,1]).
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    (Original post by gemma331)
    Let (P([0,1]),||\cdot ||_{\infty}) be the space of polynomials on [0,1] with the supremum norm. Show that (P([0,1]),||\cdot ||_{\infty}) is not complete
    As Mark85 says, you might have seen results called the Weierstass Approximation theorem, or the Stone-Weierstass theorem, which tell you that the space of polynomials is dense in the space of continuous functions - this answers your problem straight away.

    If you want to find an explicit example of a sequence of polynomials which is Cauchy but doesn't converge to a polynomial, my advice would be to think about a particular power series and its truncations.
 
 
 
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