Subset of normed Linear Space is closed The problem in question is as follows:
"Show that the set $P$ of all polynomials on the segment $[a,b]$ is a linear space. For $P$ considered as a subset of the normed linear space $C_[a,b]$ with the norm 
$||f(x)||$ = $max_{a{\leq}x{\leq}b}$ $|f(x)|$
Show that $P$ fails to be closed. "
The first part of this question is trivial as we can just show linearity through addition, scalar multiplication and distributivity. 
However, I'm having a hard time understanding how the norm gives me information about the closure of the set. Perhaps I'm thinking about closure wrong as it relates to normed spaces wrong but any help would be appreciated

Are you familiar with the Weierstrass Approximation theorem?
http://www.mast.queensu.ca/~speicher/Section14.pdf
This pdf does a good job explaining it. But the statement of the theorem should be enough alone, for your purposes: there are continuous functions which are not polynomials, but by the theorem, are the uniform limit of polynomials.
