Fourier Series of Finite Power Periodic Signals

Let us consider the Hilbert space ℓ_ℂ2 of complex sequences a = {a _n }, n ∈ ℤ, such that $${\sum\nolimits_{n \in \mathbb{Z}} {|{a_n}|} ^2} \prec \infty $$ with the Hermitian product (43)$${\left\langle {a,b} \right\rangle _{l_\mathbb{C}^2}} = \sum\limits_{n \in \mathbb{Z}} {{a_n}b_n^*} $$ and the Hilbert space L_ℂ2 ([0, T], dt/T) of complex signals x = {x(t)}, t ∈ ℝ, such that $$\int_0^T {{{\left| {x(t)} \right|}^2}dt} < \infty $$ , with the Hermitian product (44)$${\left\langle {x,y} \right\rangle _{L_\mathbb{C}^2\left( {[0,T],\frac{{dt}}{T}} \right)}} = \int_0^T {x(t)y{{(t)}^*}\frac{{dt}}{T}} $$