Inequality of fourier coefficients

Question

Rudin book's, page 101, chapter 5.

My question is: why does the inequality holds in $\text{(6)}$?

Thank you all.

Answer 1

Just estimate: $$\begin{align*} ||\Lambda_{n}||&=\sup_{\{f:||f||_{\infty}=1\}}|\Lambda_{n}f|\\ &=\sup_{\{f:||f||_{\infty}=1\}}\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)D_{n}(-t)\;dt\right|\\ &\leq\sup_{\{f:||f||_{\infty}=1\}}\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(-t)||D_{n}(t)|\;dt\\ &\leq\sup_{\{f:||f||_{\infty}=1\}}||f||_{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}|D_{n}(t)|\;dt\\ &=\frac{1}{2\pi}\int_{-\pi}^{\pi}|D_{n}(t)|\;dt\\ &=||D_{n}||_{1}. \end{align*} $$