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 $K$?