On a measure space X, the set of square integrable L^2-functions is an L^2-space. Taken together with the L^2-inner product with respect to a measure μ, 〈f, g〉 = integral_X f g d μ the L^2-space forms a Hilbert space. The functions in an L^2-space satisfy 〈ϕ|ψ〉 congruent integral ψ^_ ϕ d x and (〈ϕ|ψ〉)^_ = 〈ψ|ϕ〉 〈ϕ|λ_1 ψ_1 + λ_2 ψ_2〉 = λ_1 〈ϕ|ψ_1 〉 + λ_2 〈ϕ|ψ_2 〉 〈λ_1 ϕ_1 + λ_2 ϕ_2|ψ〉 = λ^__1 〈ϕ_1|ψ〉 + λ^__2 〈ϕ_2|ψ〉 〈ψ|ψ〉 element R>=0 ( left double bracketing bar 〈ψ_1 |ψ_2〉 right double bracketing bar )^2<=〈ψ_1 |ψ_1〉〈ψ_2 |ψ_2〉.

completion | Hilbert space | L^2-function | L^2-inner product | L^2-norm | Lebesgue integral | Lebesgue measure | L^p-space | measure | measure space | Riesz-Fischer theorem | Schwarz's inequality