We denote by $\mathcal{C}([-1,1])$ the vector space of continuous functions from $[-1,1]$ to $\mathbb{C}$ and $\mathcal{T}([-1,1])$ the complex vector subspace of $\mathcal{C}([-1,1])$ generated by the functions
$$e_k : t \mapsto e^{i\pi k t}, \quad k \in \mathbb{Z}.$$
Show that $\mathcal{T}([0,1])$ is a subalgebra of $\mathcal{C}([-1,1])$ for the usual multiplication law of functions.