# Pointwise convergence if and only if $\|f_n-f\| \to 0$

Is there a norm in the C[0,1] space function such that this happens?

$f_n(x) \rightarrow f(x)$ if and only if $\|f_n - f\| \rightarrow 0$

Whene $f_n$ is a function sucession, and $f_n(x) \rightarrow f(x)$ means Pointwise convergence

-
A norm on what space of functions? And is $f_n(n)$ a typo? –  Nate Eldredge Sep 19 '12 at 1:16
Assuming you mean $f_n(x)$ rather than $f(x)$, this is true by definition, in the sense that $f_n\to f$ in the norm topology induced by $\|\cdot\|$. To make the question meaningful you need to specify the topology in which you want $f_n\to f$. –  Alex Becker Sep 19 '12 at 1:18
"Puntal convergence" is very amusing, and practically a googlewhack to boot. It sure sounds like you mean "pointwise", so I edited it to that. –  rschwieb Sep 19 '12 at 1:22
@rschwieb en.wikipedia.org/wiki/Puntal so now what is a googlewhack? Meanwhile a punt is a type of boat or a method of kicking a ball. Pointwise also works. –  Will Jagy Sep 19 '12 at 1:24
Take a look at this: math.stackexchange.com/q/33476/9464 –  Jack Sep 19 '12 at 1:33

Suppose there were such a norm; call it $\|\cdot\|$. For each $n$, let $f_n$ be any nonzero continuous function supported in $(0,1/n)$. Since $f_n$ is not the zero function, we must have $\|f_n\| > 0$. So if we let $g_n = \frac{1}{\|f_n\|} f_n$, then we have $\|g_n\| = 1$ for all $n$; in particular, $g_n$ does not converge to zero in the norm $\|\cdot\|$. But $g_n$ is a continuous function supported in $(0,1/n)$, so $g_n(x) \to 0$ pointwise. This is a contradiction.