Convergence of sequence of functions $\{f_n\}$ to $f$ in $L^p$ if and only if $\|f_n\| \to \|f\|$

Question

Let $\{f_n\}$ be a sequence of functions in $L^p( [0,1])$, $1 \leq p < \infty$, which converges almost everywhere to a function $f$ in $L^p$. Show that $\{f_n\}$ converges to $f$ in $L^p$ norm if and only if $\|f_n\| \to \|f\|$.

What does $\|f_n\| \to \|f\|$ mean and which convergence theorem is this? I know $\{f_n\}$ converges to $f$ in $L^p$ norm means

$$ \lim_{n \to \infty} \|f_n - f\|_p = 0 $$

Answer 1

It means that $\|f_n\|_p \to \|f\|_p$ in $\Bbb R$.

Answer 2

Hints for a proof: Let $\|f_n\|_p \to \|f\|_p$. Now $2^{p}(|f|^{p}+|f_n|^{p} -|f_n-f|^{p})$ is non-negative and an application of Fatou's Lemma to this gives $\lim \sup \int |f_n-f|^{p} =0$. The converse part is true for any norm.