Moving limit into sup norm

Question

I need help understanding where I'm going wrong with this line of thought:

Assume $f_n$ converges pointwise to $f$, so $\lim \limits_{n \rightarrow \infty}f_n(x) = f(x) \forall x \in X$, then since the suprumum norm is a norm therefore continuous we can move a limit inside, like this:

$\lim \limits_{n \rightarrow \infty}\|f_n-f\|_\infty = \|\lim \limits_{n \rightarrow \infty}(f_n-f)\|_\infty = \|0\|_\infty = 0$

So all pointwise convergent sequences are also uniform convergent.

This is clearly not the case so where am I going wrong?

Answer 1

The norm is only necessarily continuous with respect to the topology that it induces. You need to be careful since you have two different topologies present here - the topology of pointwise convergence and the topology induced by $\| \cdot \|_\infty$ (which is the topology of uniform convergence).

As a result, in order to take a limit inside the norm, you need to have convergence in the topology for $\| \cdot \|_\infty$. That is, you need to have $f_n \to f$ uniformly and not just pointwise. Then the fact that you can take the limit inside the norm as in the question is just the trivial statement that uniform convergence implies uniform convergence.

Answer 2

The norm in this case is continuous in the sense that $u_{n}\rightarrow u$ in $L^{\infty}$, then $\|u_{n}\|_{L^{\infty}}\rightarrow\|u\|_{L^{\infty}}$, or we can write $\lim_{n}\|u_{n}\|_{L^{\infty}}=\|\lim_{n}u_{n}~(\text{limit taken in }L^{\infty})\|_{L^{\infty}}=\|u\|_{L^{\infty}}$.