I'm trying to prove (or disprove) that in the set $C[0,1]$ of continuous (bounded) functions on the real interval [0,1] with the integral norm $\|f(x)\|_1 = \int_0^1|f(x)|dx$ that a sequence of functions $f_n$ is convergent to $f$ if and only if $f_n$ is pointwise-convergent to $f$.
My intuition tells me that this is true, but I'm having trouble formulating an attack (particularly for the convergence => pointwise convergence case).
Given $\forall \epsilon > 0 \; \exists N \in \mathbb{N} \;$
$\forall n \geq N \; \int_0^1 |f_n(x) – f(x)|dx < \epsilon $,
I'm trying to prove that for an arbitrary $x \in [0,1]$, $\forall \epsilon_0 > 0$ there is an $N_0$ such that for $n_0 \geq N_0 |f_{n_0}(x) – f(x)| < \epsilon_0$. I think I want to make $N_0$ somehow depend on the that integral (or a 'slice' of it), but I don't see it yet. Any pointers/advice/hints are welcome.
Best Answer
It is possible for a sequence of continuous functions to converge in the $1$-norm to a continuous function, while converging pointwise nowhere.
It is also possible for a sequence of continuous functions to converge pointwise everywhere to a continuous function without converging in $1$-norm.
The last claim is the easier of the 2 to verify. Consider functions whose graphs are spikes of rapidly increasing height with base on say $[0,1/n]$.
For the first claim, you can consider a more complicated sequence of spikes, again getting narrower, but this time not increasing in height, and this time have them move all over the place (but in an orderly enough fashion to ensure the lack of convergence at any point).