Weak convergence problem Let $\Omega$ ∈ $\mathbb{R}^d$ be a bounded set and $d\ge$ 1. Consider function sequence $f_n ∈ L_3(\Omega)$ such that,
$$f_n \to f\mbox{ weakly in }  L_2(\Omega)\mbox{ and }  \|f_n\|_{L_3(\Omega)} \le M$$
for set constant $M$. We additionally know that,
$$ \|f_n\|_{L_2 (\Omega)} \to \|f\|_{L_2 (\Omega)} $$
Show that,
$$ \|f_n - f \|_{L_p(\Omega)} \to 0\mbox{ for  every } p \in [2,3)   $$
Whether limit function $f$ have to be an element of $ L_3(\Omega) $ ?
Thank for help.
 A: It's well known that norm convergence in $L^2$ follows from weak convergence plus the convergence of the norms. This gives that $f_n\to f$ pointwise a.e. on a subsequence, and thus $f\in L^3$ by Fatou's Lemma.
Now
$$
\int_{|f_n-f|\le C} |f_n-f|^p \le C^{p-2}\|f_n-f\|_2^2
$$
and
$$
\int_{|f_n-f|>C} |f_n-f|^p \lesssim C^{p-3} ,
$$
since at least one of $f_n$, $f$ is $>C/2$ on this set and, as we observed, the $3$ norms are bounded. So if $2\le p<3$, then we can make $\|f_n-f\|_p$ arbitrarily small by taking both $C$ and $n$ large.
A: First show the strong $L_2$ convergence. This follows from 
$$
\|f_n - f\|^2_2 = \|f_n\|^2_2 - 2 \langle f,f_n \rangle + \|f\|_2^2 \, .
$$
Just take the limit of each term, using your assumptions. 
The $L_p$ convergence for all $p \in [2,3)$ now follows from Hölder's inequality.   
Since the sequence converges also weakly in $L^3$, the limit is indeed in $L^3$. 
Added in response to the comments.
To prove convergence in $L^p$ with $2 < p < 3$, write $p = 2+r$ with $0 < r < 1$ and use Hölder's inequality: 
$$
\|f_n - f_m\|_p^p = \int_\Omega |f_n-f_m|^{2(1-r)}|f_n-f_m|^{3r}  \le \left(\int_\Omega|f_n-f_m|^2\right)^{1-r}  
\left(\int_\Omega|f_n-f_m|^3\right)^r \, .
$$
Since $\|f_n - f_m\|_2 \to 0$ as $m, n \to \infty$ and $\|f_n - f_m\|_3$ remains bounded, this shows that the sequence in Cauchy in $L^p$ and thus converges strongly. The limit must agree with the $L^2$ limit. 
To prove weak convergence in $L^3$, use test functions in $L^2 \cap L^{4/3}$, e.g. smooth compactly supported test functions (as suggested in the comments), plus the fact that the $f_n$ are bounded set in $L^3$.   
