There is it: Is it true for every $f\in C(T)$ that the Fourier series of $f$ converges to $f(x)$ at every point $x$?
Let us recall that the $n$th partial sum of the Fourier series of $f$ at the point $x$ is given by
$$s_n(f;x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)D_n(x-t)\,dt\quad(n = 0,1,2,\cdots)\quad\quad (1)$$.
where $$D_n(t)=\sum_{k = -n}^n e^{ikt}$$ The problem is to determine whether $$\lim_{n\to\infty}s_n(f;x)=f(x)$$ for every $f\in C(T)$ and for every real $x$.
We shall see that the Banach-Steinhaus theorem answers the question negatively.
Put
$$s^{*}(f;x)=\sup |s_n(f;x)|\quad(n\in N)$$
To begin with, take $x$ $=$ $0$, and define
$$\Lambda_n f= s_n(f;0)\quad(f \in C(T); n= 1,2,3,…).$$
We know that $C(T)$ is a Banach space, relative to the supremum norm $\|f\|_{\infty}$.
It follows from $(1)$ that each $\Lambda_n$ is a bounded linear functional on $C(T)$, of norm
$$||\Lambda_n|| \leq \frac {1}{2\pi}\int_{-\pi}^{\pi} |D_n(t)| dt.$$
I don’t understand why are each $\Lambda_n$ a bounded linear functional on $C(T)$, Hence I also don’t understand why we have this inequality:
$$||\Lambda_n|| \leq \frac {1}{2\pi} \int_{-\pi}^{\pi} |D_n(t)| dt?$$
Any help would be appreciated.
By definition, $\Lambda_n : C(T) \to \mathbb{R}$ is defined by $\Lambda_n(f) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) D_n(t)dt$ for each natural number $n$. This is clearly linear and $$\begin{align*} |\Lambda_n(f)| = \frac{1}{2\pi}\left| \int_{-\pi}^{\pi} f(t) D_n(t)dt \right| &\le \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(t)| |D_n(t)|dt \\ &\le ||f||_{\infty} \left(\frac{1}{2\pi} \int_{-\pi}^{\pi} |D_n(t)|dt \right), \end{align*}$$ which implies $||\Lambda_n|| = \sup_{f \ne 0} \frac{|\Lambda_n(f)|}{||f||_{\infty}} \le \frac{1}{2\pi} \int_{-\pi}^{\pi} |D_n(t)|dt < \infty$. This shows that $\Lambda_n$ is a bounded linear functional on $C(T)$. You can even further show that $||\Lambda_n|| = \frac{1}{2\pi} \int_{-\pi}^{\pi} |D_n(t)|dt$ by explicitly constructing a function for which the equality holds. Consider $g_n(x) = \begin{cases} 1 &, D_n(x) \ge 0 \\ -1 &, D_n(x) < 0 \end{cases}$, then observe that $\Lambda_n(g_n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} |D_n(t)|dt$ but sadly $g_n$ is not a continuous function. Now we can approximate $g_n$ by a sequence of continuous functions $\{ f_m\}_{m=1}^{\infty}$.