Hilbert Spaces and Orthogonal Projection in Infinite Dimensions

An inner product <x,y> generalizes the dot product: it is linear in x, obeys <y,x>=conjugate(<x,y>), and <x,x>>=0 with equality only for x=0. Its norm is ||x||=sqrt(<x,x>); <x,y>=0 means orthogonal. A sequence is Cauchy if, for every epsilon>0, ||x_n-x_m||<epsilon for all sufficiently large n,m. A Hilbert space is an inner-product space complete in this norm: every Cauchy sequence converges in norm to a vector in the space.

An orthonormal family has unit vectors with <e_i,e_j>=0 for distinct indices. An orthonormal basis is complete: its finite linear combinations are dense. Using inner products linear in the first argument, c_n=<v,e_n> and v=sum c_n e_n in norm. Parseval gives ||v||^2=sum |c_n|^2. To find one missing coefficient's size, subtract the known squared sizes from ||v||^2 and take the square root.

Riesz representation applies to a real or complex Hilbert space. Every bounded linear functional f has a unique vector y with f(x)=<x,y> for all x, and ||f||=||y||. We use inner products linear in the first argument and conjugate-linear in the second. Thus a fixed y gives a linear map of x. For real coordinates, f(x)=a dot x is represented by a.

Closed subspaces contain their convergent sequence limits. Let d=inf{||v-m||:m in M} and take m_n in M with ||v-m_n|| tending to d. Expanding inner products gives ||a+b||^2+||a-b||^2=2||a||^2+2||b||^2, the parallelogram identity. Here it yields ||m_n-m_k||^2<=2||v-m_n||^2+2||v-m_k||^2-4d^2, so m_n is Cauchy. Completeness gives its ambient limit; closedness puts it in M. A non-closed subspace may still have a closest point for a particular vector.

For an orthonormal series, square summability makes partial sums Cauchy: squared tail norms are sums of |c_n|^2. In a Hilbert space they converge in norm. With c_n=<v,e_n>, a complete family recovers every v; an incomplete one need not. c00 consists of sequences with finitely many nonzero entries, using the l2 norm. Standard e_n has entry 1 at n and 0 elsewhere. Then s_N=sum_{n<=N}(1/n)e_n is Cauchy, but its infinite-support limit is outside c00.

To check Riesz uniqueness, suppose <x,y>=<x,z> for every x. Then <x,y-z>=0 for every x. Choosing x=y-z gives ||y-z||^2=0, so y=z. In a complex space, the representing vector of a times f is conjugate(a)y, because the second inner-product argument is conjugate-linear. The representing vector's norm equals the functional norm.