$\Vert f_n\Vert_p \longrightarrow \Vert f\Vert_p$ implies the convergence of $f_n$ to $f$ in $\textit{L}^p$ with the right hypotesis

Question

I think this might work, but how i don't know:

Let $p\in (1,+\infty)$ and $\{f_n\}$, $f\in \textit{L}^p(E)$, if we know that:

• $\Vert f_n\Vert_p \longrightarrow \Vert f\Vert_p$ as $n\longrightarrow +\infty$;
• $\forall$ measurable $F\subseteq E$ with $|F| < +\infty \Longrightarrow \int_Ff_n\longrightarrow \int_Ff$, as $n\rightarrow +\infty$;

is it true that $f_n$ converges to $f$ in $\textit{L}^p(E)$ in norm (strong convergence)?

Any suggestions?

Answer 1

Here is a sketch for a proof:

• The sequence is bounded, hence it has a weakly convergent subsequence.
• The second bullet gives that the weak limit is $f$.
• A subsequence-subsequence argument shows that the entire sequences converges weakly.
• Weak convergence and convergence of norms implies strong convergence in $L^p$, $1 < p <\infty$.