Is it true that $C[0,1] \subseteq \overline {\text {span} \left \{e^{2\pi int}\ |\ n \in \mathbb Z, t \in [0,1] \right \}}\ $?

Question

Can we say that $C[0,1] \subseteq \overline {\text {span} \left \{e^{2 \pi int}\ |\ n \in \mathbb Z, t \in [0,1] \right \}}\ $?

I think that it's true since any continuous function on $[0,1]$ admit Fourier series expansion. But I am not comfortable with such things. Can anybody confirm whether it holds good or not?

Thanks for your time.

Answer 1

Every function on the right side satisfies $f(0)=f(1)$. So the statement is false (if you are using the usual sup norm on $C[0,1]$).

However continuous funcions $f$ satisfying $f(0)=f(1)$ do belong to the right side. This is an immediate consequence of Fejer's Theorem. See https://en.wikipedia.org/wiki/Fej%C3%A9r%27s_theorem