The product of square integrable functions is the space of square integrable functions on the product
Theorem : Given two separable Hilbert spaces H1=L2(X,μ) and H2=L2(Y,λ), there exists a unique isometric isomorphism
L2(X,μ)⊗L2(Y,λ)≅L2(X×Y,μ×λ)which maps f⊗g to f(x)g(y).
Proof : Being separable Hilbert spaces, H1 and H2 can be given a basis, {ψ1i(x)} and {ψ2i(y)}. We need to show first that {ψ1i(x)ψ2j(y)} is a basis of L2(X×Y,μ×λ). This is true if this set obeys Parseval's identity : for any v∈L2(X×Y,μ×λ),
∑i,j|⟨v,ψ1iψ2j⟩|2=⟨v,v⟩Consider our function v(x,y). Then :
⟨v,ψ1iψ2j⟩=∫v∗(x,y)ψ1i(x)ψ2j(y)dμ(x)dλ(y)