Can anybody give an example that for $1 \leq p < \infty$ neither $\mathcal L^p (\mathbb R) \subseteq \mathcal L^{\infty} (\mathbb R)$ nor $\mathcal L^{\infty} (\mathbb R) \subseteq \mathcal L^p (\mathbb R)$ holds?
And why is for $f \in C([0,1], \mathbb C): ||f||_{\infty} := sup \{ |f(x)|: x \in [0,1] \} = ||f||_{L^{\infty}}$?
Any help is appreciated.
The standard example for showing that $L^{\infty}(\Bbb{R})\not\subseteq L^p(\Bbb{R})$ would be $f(x) = 1$ for all $x\in\Bbb{R}$.
An example in the other way would be the function
$$f(x) = \begin{cases} n & x\in[n,n+\frac{1}{n^3}] \\ 0 & \text{otherwise}\end{cases}$$
here, $n$ is a natural number. The areas of each rectangular piece of $f$ are $\frac{1}{n^2}$ so $f$ will have finite integral, giving that $f\in L^1(\Bbb{R})$ but clearly $f\not\in L^{\infty}(\Bbb{R})$ so $L^1(\Bbb{R})\not\subseteq L^{\infty}(\Bbb{R})$. A suitable adjustment shows that $L^p(\Bbb{R})\not\subseteq L^{\infty}(\Bbb{R})$.