Prove. Pythagorean theorem in Hilbert space. Let f ₁(x), f ₂(x), .... , f_N(x) be N pairwise orthogonal functions and let

            f(x) = f ₁(x) + f ₂(x) + .... + f_N(x)

be their sum. Then the square of the length of f is equal to the sum of the squares of the lengths of f ₁(x), f ₂(x), .... , f_N(x). Because the lengths of vectors in Hilbert space are given as integrals, the Pythagorean theorem in Hilbert space takes the form

Proof. The proof follows directly from the properties of the inner (dot) product, remembering that f_i•f_j = 0 when i j. For example, consider the case for three-dimensional space. Let f (x) = f₁ + f₂ + f₃. We wish to prove (f₁ + f₂ + f₃)•(f₁ + f₂ + f₃) = f₁•f₁ + f₂•f₂ + f₃•f₃ .

            (f₁ + f₂ + f₃)•(f₁ + f₂ + f₃) = f₁•(f₁ + f₂ + f₃) + f₂• (f₁ + f₂ + f₃) + f₃•(f₁ + f₂ + f₃)

            = f₁•f₁ + f₁•f₂ + f₁•f₃ + f₂•f₁ + f₂•f₂ + f₂•f₃ + f₃•f₁ + f₃•f₂ + f₃•f₃

            = f₁•f₁ + f₂•f₂ + f₃•f₃

From this result, 1) follows directly from the definition of the inner product in Hilbert space.