Question on 'taking out' pointwise limit in the $L^p$ norm

Question

In functional analysis, many properties of certain spaces are normally derived from taking a pointwise limit out of the norm, i.e.

$\lvert \lvert x \rvert \rvert=\lim\limits_{n\to \infty}\lvert \lvert x_{n} \rvert \rvert$ $(*)$.

The normal justification for this is that

$\lvert \lvert \cdot \rvert \rvert: X \to \mathbb R$ is a continuous function wrt to the norm $\lvert \lvert \cdot \rvert \rvert$ by the reverse triangle inequality.

Note that we merely used the definition of a norm to obtain continuity. In particular, $\lvert \lvert \cdot \rvert \rvert_{L^{p}}$ is continuous.

However, when we arrive at $L^{p}$ spaces there are convergence theorems, like Dominated Convergence Theorem, Monotone Convergence Theorem and Fatou's Lemma which of course imply that simply taking the limit out of the norm is not possible. Why is this not a contradiction to $(*)$?

Answer 1

The difference is that in the first case you assume $x_n \to x$ in the topology of the norm $\|\cdot\|$ (this means $\|x_n-x\| \to 0$) and then you conclude $\|x_n\| \to \|x\|$ by continuity of the norm.

In the context of convergence theorems in $L^p$ you assume $f_n \to f$ a.e. (this is different from $\|f_n-f\|_{L^p} \to 0$) and then conclude $\|f_n\|_{L^p} \to \|f\|_{L^p}$.