Set of Functions on Circle forms Vector Space

the set of functions $\map f \phi$ on the circle (0 \le \phi < 2 \pi) forms a vector space:
 * $\ds \map f \phi = \sum_{m \mathop = -\infty}^{+\infty} a_m e^{i m \phi}$

for integer m


 * : Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE: Example $4$