Can I ask a homework question here?
Assume that $f \in L^q(\mathbb R^d)$ for some $ q < \infty$ . show that
$\mathrm{lim}_{p \to \infty}||f||_p = ||f||_{\infty}$
$p$ conjugate of $q$
Can I ask a homework question here?
Assume that $f \in L^q(\mathbb R^d)$ for some $ q < \infty$ . show that
$\mathrm{lim}_{p \to \infty}||f||_p = ||f||_{\infty}$
$p$ conjugate of $q$
In order your question to be correct you need to require that for some $q$ we have $f\in L^p(\mathbb{R}^d)$ for all $p\geq q$. To solve this problem use the following approach
1) Take $\{p_n:n\in\mathbb{N}\}$ such that $\lim\limits_{n\to\infty} p_n=+\infty$ and $\lim\limits_{n\to\infty}\Vert f\Vert_{p_n}=\liminf\limits_{p\to\infty}\Vert f\Vert_p:=a$.
2) Using Chebyshev inequality prove that $\lim\limits_{n\to\infty}\mu(\{x\in\mathbb{R}^d:|f(x)|> a+\varepsilon\})=0$ for all $\varepsilon>0$.
3) Conclude that $\Vert f\Vert_\infty\leq a$.
4) Prove that $\Vert f\Vert_p\leq \Vert f\Vert_q^{q/p}\Vert f\Vert_\infty^{1-q/p}$ for all $p> q$ and then conclude $\limsup\limits_{p\to\infty}\Vert f\Vert_p\leq\Vert f\Vert_\infty$.
5) The rest is clear.
Let $\mu:=|f|^qd\lambda$.
This solves the question under the hypothesis that $\|f\|_q$ is finite for some fixed positive finite $q$.
Assume without loss of generality that $\|f\|_\infty=1$. Then:
Thus, for every $u\lt1$, $u\leqslant\liminf\limits_{p\to\infty}\|f\|_p\leqslant\limsup\limits_{p\to\infty}\|f\|_p\leqslant1$, hence $\lim\limits_{p\to\infty}\|f\|_p=1$.
Now, show that the without loss of generality step at the beginning of the post does not make all the rest a proof by intimidation.