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
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
No.
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.