$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$
The symbol $\sim$ has the following meaning:
We know that the right part of $(*)$ converges with the meaning of the induced norm to an element of the space $E$. We don't know by now if the series at the right part converges pointwise to $f$.
What does the following part mean?? $$\text{ ...converges with the meaning of the induced norm ... }$$
$$$$
Furthermore, the series at the right part is periodic with period $L$.
How do we know that the period is $L$??
The induced norm should reflect the special norm derived from the usual inner product on $L_2$.
The second part can be derived using the properties of sine and cosine functions.