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Consider the linear space of polynomials on [$a$, $b$] normed by $\rVert$\centerdot$\rVert$_max norm. Is this normed linear space a Banach space? My professor said it is not, but then could I use contradiction? Haven't been able to come up with a good Cauchy sequence argument, though. I would appreciate all the help I could get, thank you.

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    see my answer for a counterexample.2011-04-15

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Approximate the exponential function by its Taylor polynomials. You should also prove that the exponential function is not a polynomial, not even in an interval.

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    I am working on the same problem and was wondering about this. Does the argument work like this: The taylor series is convergent (in what space?) => the Taylor series is cauchy. Since it is made up of polynomials it is a cauchy sequence in P[a,b] but is not convergent => P[a,b] not complete. Does that look OK?2015-12-13
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I think I remember using Baire Category to do this. Let $A_n$ be the space of polynomials of degree less than n and clearly the space of linear polynomials is a countable union of the $A_n$. Then prove this is closed but does not contain an interval. I think proving it is closed is the hard part.

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    @user1736: A nice problem is: if $f: (0,\infty) \to \mathbb{R}$ is continuous with the property that $\lim_{n \to \infty} f(nx) = 0$ for all $x \in (0,\infty)$ then $\lim_{x \to \infty} f(x) = 0$. The analogous result doesn't hold for $f(n + x)$. You can find a number of such exercises in any book on functional analysis, mostly in connection with the fundamental theorems (uniform boundedness, open mapping, etc.) Another of my favourites: show that $[0,1]$ is not the union of a countably many non-empty, pairwise disjoint closed sets.2011-04-14