# Properties of Norm spaces

Suppose $$m(E)<\infty$$ and $$f\in\mathcal{L}^{\infty}(E)$$. The goal of this problem is to show \begin{align*} \lim_{p\to \infty}\|f\|_p=\|f\|_{\infty}. \end{align*} First, prove that \begin{align*} \lim_{p\to\infty}\|f\|_p\leq \|f\|_{\infty}. \end{align*}Proof: Assume $$m(E)<\infty$$ and $$f\in\mathcal{L}^{\infty}(E)$$. By definition, \begin{align*} \|f\|_p=\left(\int_E|f|^pdm\right)^{1/p}\leq \left(\int_E\|f\|_{\infty}^p\right)^{1/p}=\|f\|_{\infty}\cdot m(E)^{1/p}<\infty. \end{align*} Letting $$p\to \infty$$, we have that $$m(E)^{1/p}\to 1$$. Therefore, \begin{align*} \lim_{p\to\infty}\|f\|_p\leq \|f\|_{\infty}. \end{align*}

I'm not sure if I am missing anything here or need to justify being able to bring the limit inside and applying to $$m(E)$$.

Next, I need to prove that \begin{align*} \lim_{p\to\infty}\|f\|_p\geq \|f\|_{\infty}-\epsilon \end{align*} for any $$\epsilon>0$$. Hint: Look at the set \begin{align*} F=\{x\in E:|f|>\|f\|_{\infty}-\epsilon\}. \end{align*}

I'm stuck on this part and would appreciate any help, thanks.

You have that

$$||f||_p^p=\int_F|f|^p+\int_{F^c}|f|^p>\int_F|f|^p >$$

$$>\int_F (||f||_\infty -\epsilon)^p=(||f||_\infty -\epsilon)^pm(F)$$

Then

$$||f||_p> (||f||_\infty -\epsilon)(m(F))^\frac{1}{p}$$

So if you fixed $$\epsilon$$ (and $$F=F_{\epsilon}$$ ) you have that for $$p\to \infty$$

$$lim_p||f||_p>||f||_\infty -\epsilon$$

for every $$\epsilon>0$$ so

$$lim_p ||f||_p>||f||_\infty$$