The trigonometric system as a Schauder basis

Prove that the system $\left\{ e^{i n t}\right\}_{-\infty }^{\infty }$ is not a Schauder basis in the space $C_{\text{per }}[0,2 \pi ]$.

Prove that the system $\left\{ e^{i n t}\right\}_{-\infty }^{\infty }$ is a Schauder basis in the spaces $L_{p}[0,2 \pi ], p \in (1, \infty )$.

Prove that the system $\left\{ e^{i n t}\right\}_{-\infty }^{\infty }$ is not a Schauder basis in the space $L_{1}[0,2 \pi ]$ (see also problem 10.29). Prove that the system $\left\{ \sin (n t)\right\}_{1}^{\infty }$ is not a Schauder basis in the spaces $L_{1}[0, \pi ]$ and

(See Chapter 14 for proof that $\left\{ e^{2 \pi i n t}\right\}_{n \in \mathbb {Z}}$ is a basis for $L^{p}(\mathbf{T})$ when $1 < p < \infty$, but is not a basis for $L^{1}(\mathbf{T})$ or $C(\mathbf{T})$.)

Prove Lemma 14.1.

Finish the proof of Theorem 14.2.

Prove that the system of functions $\left\{ e^{2\pi inx}\right\}$, $n\in \mathbb {Z}$, is not a topological basis (see Problem 410) in the space $CP[0,1]$ of all continuous functions on $[0,1]$ with the condition $f(0)=f(1)$.

Prove the stated assertion. The trigonometric system $1, \cos x, \sin x, \ldots , \cos n x, \sin n x, \ldots$ is not a basis in the space $C^{*}[-\pi , \pi ]$ (see problem 115).