§ 2.2. Functional analysis and topology.
Physics describes objects by functions on appropriate domains. Functions on a manifold must respect its topology.
Example 1c: Functional analysis on the circle. We model the circle by the interval on the real line. Consider on all exponentiaL functions
Their values will be unique on the domain provided they are periodic, that is
Periodicity requires , and so restricts the values of to the discrete subset . The discrete subset of functions on the interval fulfills the homotopic boundary condition .
Repetition of the discrete translation step eq.10 generates the discrete translation or infinite cyclic group . The periodic set is the basis of complex Fourier series analysis of a periodic function. It represents any periodic function as an infinite series with complex coefficients . These can be found by integration over the interval :
We conclude that the particular functions form a basis of all functions on the circle and, when extended from the interval to the real line, of all periodic functions on the line.
A useful and constructive characterization of the basis of Fourier analysis arises in terms of the deck transformation group: Any basis function is invariant under the action of the discrete translation group , which in turn is isomorphic to the fundamental group of the circle. In case of the circle it suffices already to require the invariance under the generator of the translation group, that is to require periodicity eq. 11.