I am working with the following map:
Let $(E,\mathcal{A},\mu)$ be a measure space. $p\in[1,\infty)$. Define $$\Phi:\begin{cases} L^p(\mu)\to L^1(\mu)\\ f \to |f|^p \end{cases} $$ How could we show that this map is continuous? I was trying to write $\|\Phi(f+h) - \Phi(f)\|_1 = \int \left||f+h|^p - |f|^p\right| d\mu$ and I am stuck here.
On
Here is a proof assuming that $\mu(E)<\infty.$ If $p=1,$ there is nothing to do. Otherwise, from the calculation
$|a+b|^p-|a|^p=p\int_0^1|a + tb|^{p-2}(a + tb)b dt\Rightarrow ||a+b|^p-|a|^p|\le p\int_0^1(|a| + |b|)^{p-1}|b| dt\le p2^{p-1}(|a|^{p-1}+|b|^{p-1})|b|=p2^{p-1}|a|^{p-1}|b|+p2^{p-1}|b|^p,$
the result follows by taking $a=f(x),\ b=h(x)$, integrating and noting that $h\in L^1(\mu)$ and $f\in L^{p-1}$ because $\mu(E)<\infty.$
If $f_n \to f$ in $L^{p}$ then $\|f_n\|_p \to \|f\|_p$. $\|f_n\|_p^{p} \to \|f\|_p^{p}$ or $\int |f_n|^{p} \to \int |f|^{p}$. Now go to subsequences and use the following:
If $g_n \to g$ a.e. , $g_n \geq 0$ a.e. and $\int g_n \to \int g$ then $\int|g_n-g| \to 0$.
[This is called Scheffe's Lemma].
Hence every subsequence of $(|f_n|^{p})$ has a subsequence which converges in $L^{1}$ to $|f|^{P}$