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

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!

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 $\|f_n\|_p\to \|f\|_p$.

• thanks! this looks very simple. But is my answer wrong? Jul 11 '17 at 5:06
• The theorem in Royden proves a stronger result (if and only if) because it assumes pointwise convergence. For the direction you want, pointwise convergence is not needed. Jul 11 '17 at 5:09
• But we get pointwise convergence from R-F Jul 11 '17 at 5:11
• Or rather p.w convergence of a subsequence. Nevermind. Thank you for your help Jul 11 '17 at 5:12
• Well, then using Royden's theorem as a black box would only give you the convergence you want for a subsequence. In any case, my guess is that Royden's proof of the direction you want is the same as my proof. It's actually the other direction that's more interesting. Jul 11 '17 at 5:14

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

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.