Use inverse mapping theorem to prove that C[0,1] with Lp norm is incomplete I'm reading Bollabas's Linear Analysis, and here's a question in Chap5 (pp82, q13):

Show that for $1 \leq p < \infty$ the $L_p$ norm ($||f||_p :=
> (\int_0^1|f(t)|^pdt)^{\frac{1}{p}})$ on $C[0,1]$ is dominated by the
uniform norm $||f|| = \sup_{0\leq t \leq1} |f(t)|$, and deduce that
$C[0,1]$ is incomplete in the $L_p$ norm. (Hint: use inverse mapping
theorem)

I think the proof without using the hint is not hard, just construct a sequence of function $f_n$ that converge to a "step" function which is not continuous. But I'm curious how to prove this theorem by leveraging theses two hints: 1) dominated by $L_\infty$, 2) inverse mapping theorem

Let $X = (C[0,1],\|\cdot\|_\infty)$ and let $Y = (C[0,1],\|\cdot\|_p)$. The identity map $I:X \to Y$ is bounded since $$\|If\|_p = \|f\|_p \le \|f\|_\infty.$$
If $X$ and $Y$ both happened to be Banach spaces you'd have that the inverse is bounded too. The inverse of the identity is itself, so this would imply in turn that there is a constant $C$ satisfying $$\|f\|_\infty \le C \|f\|_p.$$
There are lots of examples to show this isn't the case. It is easy to write down a sequence $\{f_n\}$ of continuous functions for which $\|f_n\|_p$ is bounded yet $\|f_n\|_\infty \to \infty$.
The conclusion is that, since $X$ is a Banach space, $Y$ is not.
