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

Question

Suppose $(X,M,\mu)$ is a measure space, $\mu$ a positive measure $f_n,f\in L^{p}(X)$ and $1\leq p < \infty$. I want to prove that

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

I don't really know how to prove this but I have some ideas.

By the Riesz-Fischer theorem, $\{f_n\}\to f$ in $L^p(X)$ implies that there exists a subsequence of $\{f_n\}$ that converges pointwise a.e to $f$ on $X$. And there is another theorem in Royden's which states that:

Theorem 7 Let $E$ be a measurable set and $1\leq p<\infty$. Suppose $\{f_n\}$ is a sequence in $L^p(E)$ that converges pointwise a.e. on $E$ to the function $f$ which belongs to $L^p(E)$. Then $$\{f_n\}\to f\ \text{in}\ L^p(E)\ \text{if and only}\ \lim_{n\to\infty}\int_E|f_n|^p=\int_E|f|^p.$$

Does this imply the statement is true? Any help is appreciated. Thank you!

Answer 1

For any normed vector space $(V,\|\cdot\|)$, it follows from the triangle inequality that $$ \lvert\|x\|-\|y\|\rvert\leq \|x-y\| $$ for all $x,y\in V$.

Applying this inequality to $L^p(X)$, we have $$ \lvert \|f_n\|_p-\|f\|_p\rvert\leq \|f_n-f\|_p$$ hence if $f_n\to f$ in $L^p$ then $\|f_n\|_p\to \|f\|_p$.

Answer 2

Hint: In any normed linear space $|\|x\|-\|y\|| \le \|x-y\|.$

Answer 3

We are dealing the Normed linear space. So, please remember this method $||f_{n}||_{p}=||f_{n}-f+f||_{p}≤||f_{n}-f||_{p}+||f||_{p}.$

$\therefore$$||f_{n}||_{p}-||f||_{p}≤||f_{n}-f||_{p}.$

By the symmetry, $| ||f_{n}||_{p}-||f||_{p}|≤||f_{n}-f||_{p}.$ We can deduce from this.