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Let $C[a,b]$ be the set of all continuous functions on $[a,b]$, with the $p$-norm for $p$ in $[1,\infty]$. Let $T$ be the mapping defined by:

$$T:g \to g^2$$

where $g$ belongs to $C[a,b]$. Is this map continuous for all $p$?

EDIT: The $p$-norm is defined as $\|g\|_p = (\int_a^b \! |g(x)|^{p} \, \mathrm{d} x)^{1/p}$

EDIT 2: Would it be correct to show that:

$\|Tg-Tf\|_p \le K\|g-f\|_p$ for some constant $K$?

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Do you know how to use TeX? – Nikita Evseev Oct 14 '12 at 16:34
Unfortunately I do not. I will look over the TeX help and try to edit the question though. – Heisenberg Oct 14 '12 at 16:35
Is that better? :) – Heisenberg Oct 14 '12 at 16:38
I've just edited. – Sigur Oct 14 '12 at 16:39
Thanks Sigur. I just made the change before you did. I did not realize posting in $TeX$ was that simple. – Heisenberg Oct 14 '12 at 16:39
up vote 1 down vote accepted

This map is not continuous for any $p$.

Hint: Consider a piecewise linear function with $f(a) = c$, $f(a+r)=0$, and $f=0$ on $[a+r, b]$. Compute the $p$ norm of $f$ and $f^2$ in terms of $c,r$. Then choose a sequence of $c_n$ and $r_n$ such that for the corresponding functions $f_n$, $\|f_n\|_p \to 0$ but $\|f_n^2\|_p \to \infty$.

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