Let x ( t ) be an arbitrary function in L 2 () and define the sequence of functions x n ( t ) =

Let x(t) be an arbitrary function in L₂(ℝ) and define the sequence of functions x_n(t) = x(t + nT) for n ∈ ℤ.

a. Show that ⟨x(t + mT), x(t + nT)⟩ = ⟨x(t), x(t + (n − m)T)) for all m ∈ ℤ, where the inner product is the standard inner product on L₂. 

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b. Use the previous result to show that the functions {x_n(t)} form an orthonormal set if and only if ⟨x(t), x(t + nT)⟩ = δ_0n.