Q:

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that 
convergence pointwise  implies convergence in norm $\Vert .\Vert$ ?
I think not because if there would be the natural candidate standard $C([a,b],\mathbb{R})$
any suggestion is welcome.

A:

There is no such norm. In fact, for every norm on $C^\infty([0, 1])$ there is a
sequence of smooth functions that converges pointwise but not in the norm.
Let $\Psi$ be a bump function on $\mathbb{R}$ with support $[0,1]$ 
and define $\{f_n\}_{n=1}^\infty$ by $f_n(x) = \Psi(nx)$.
Although $f_n$ is never identically zero, it is easy to verify that for every $x$ the sequence $\{ f_n(x) \}$ is eventually
zero, so $f_n \to 0$ pointwise. 
If we fix a norm, we can define $g_n = nf_n/\|f_n\|$. Since $g_n$ is zero 
wherever $f_n$ is, we also have $g_n \to 0$ pointwise. On the other hand, 
$\|g_n\| = n$ is unbounded, so $\{g_n\}$ is not even a Cauchy sequence in the norm.

