Update: I understand the separability of $L^p(T)$ now, but I'm unable to prove the non-separability of $L^\infty(T)$. Could someone please provide some more details/hints? Thank you!
Prove that $L^p(T)$ is separable for $1 \le p < \infty$, and $L^\infty(T)$ is not separable.
$T$ is the unit circle in $\mathbb C$, i.e. $T = \{z\in\mathbb C: |z| = 1\}$. We define $L^p(T)$ for $1\le p < \infty$ as the class of all complex, Lebesgue measurable, $2\pi$-periodic functions on $\mathbb R^1$, for which the norm $$\|f\|_p = \left\{\frac{1}{2\pi}\int_{-\pi}^\pi |f(t)|^p\, dt \right\}^{1/p}$$ is finite. We define $L^\infty(T)$ similarly.
First of all, what metric do we use to talk about density here? I assume we're using the metric induced by the $L^p$ norm.
Attempt for $L^p(T)$ with $p\in [1,\infty)$:
I am trying to explicitly construct a countable dense subset of $L^p(T)$. I have a proposal for a countable dense subset, which I am sure is countable, but I'm unable to show it is dense. In case it is not dense, I would appreciate any other suggestions. I define $$\mathcal C = \bigcup_{n=1}^\infty \mathcal C_n$$ where $$\mathcal C_n = \{f: [-\pi,\pi] \to \mathbb C: f \text{ takes values in }\mathbb Q \text{ on endpoints of } I_{n,k},\\ f \text{ is linear on }I_{n,k}, 0\le k \le n-1, f(\pi) = f(-\pi)\}$$ and $$I_{n,k} = \left[-\pi + \frac{2\pi k}{n} , -\pi + \frac{2\pi (k+1)}{n}\right] \quad (0 \le k \le n-1)$$
Clearly, $$|\mathcal C_n| = |\mathbb Q^n|\quad \text{and} \quad |\mathcal C| = \left|\bigcup_{n=1}^\infty\mathbb Q^n \right|$$
so $\mathcal C$ is countable. Is it dense too?
Attempt for $L^\infty(T)$:
How do I get started with showing that $L^\infty(T)$ is not separable? I am trying a proof by contradiction. Suppose $\{f_n\}_{n=1}^\infty$ is a countable, dense subset of $L^\infty(T)$ (which can be identified with $L^\infty[-\pi,\pi]$.) The norm on $L^\infty(T)$ would be the essential supremum norm. What's next?
For $L^{\infty}(T).$
Let $(b_n)_{n\in\Bbb N}$ be a strictly increasing sequence of members of $[0,2\pi)$ with $b_1=0$ and $\lim_{n\to\infty}b_n=2\pi.$
For any $g\in \{0,1\}^{\Bbb N}$ that is, any $g:\Bbb N\to \{0,1\},$ and for any $t\in [0,2\pi)$ let $f_g(t)=2g(n)$ if $t\in [b_n,b_{n+1}).$ For unequal $g_1,g_2\in \{0,1\}^{\Bbb N}$ we have $\|f_{g_1}-f_{g_2}\|_{\infty}\ge 2.$
So the family $F= \{B_{\infty}(f_g,1):g\in\{0,1\}^{\Bbb N}\}$, of non-empty open balls of radius $1$, is pairwise-disjoint and has cardinal $|F|=|\{0,1\}^{\Bbb N}|=2^{\aleph_0}.$
If $D$ is dense in $L^{\infty}(T)$ then $D$ must intersect each member of $F$, implying $|D|\ge |F|.$