Set of Functions on Circle forms Vector Space

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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

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE: Example $4$