In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x in a Hilbert space H and every nonempty closed convex there exists a unique vector m \in C for which |c - x| is minimized over the vectors c \in C; that is, such that for every c \in C.

Finite dimensional case

Some intuition for the theorem can be obtained by considering the first order condition of the optimization problem. Consider a finite dimensional real Hilbert space H with a subspace C and a point x. If m \in C is a or of the function defined by (which is the same as the minimum point of ), then derivative must be zero at m. In matrix derivative notation Since \partial c is a vector in C that represents an arbitrary tangent direction, it follows that m - x must be orthogonal to every vector in C.

Statement

Detailed elementary proof

Proof by reduction to a special case

It suffices to prove the theorem in the case of x = 0 because the general case follows from the statement below by replacing C with C - x.

Consequences

If then which implies c = 0. Let where \mathbb{F} is the underlying scalar field of H and define which is continuous and linear because this is true of each of its coordinates The set is closed in H because { 0 } is closed in P and L : H \to P is continuous. The kernel of any linear map is a vector subspace of its domain, which is why is a vector subspace of H. Let x \in H. The Hilbert projection theorem guarantees the existence of a unique m \in C such that (or equivalently, for all ). Let p := x - m so that and it remains to show that The inequality above can be rewritten as: Because m \in C and C is a vector space, m + C = C and C = - C, which implies that The previous inequality thus becomes or equivalently, But this last statement is true if and only if every c \in C. Thus

Properties

Expression as a global minimum The statement and conclusion of the Hilbert projection theorem can be expressed in terms of global minimums of the followings functions. Their notation will also be used to simplify certain statements. Given a non-empty subset and some x \in H, define a function A of d_{C,x}, if one exists, is any point m in such that in which case is equal to the of the function d_{C, x}, which is: Effects of translations and scalings When this global minimum point m exists and is unique then denote it by \min(C, x); explicitly, the defining properties of \min(C, x) (if it exists) are: The Hilbert projection theorem guarantees that this unique minimum point exists whenever C is a non-empty closed and convex subset of a Hilbert space. However, such a minimum point can also exist in non-convex or non-closed subsets as well; for instance, just as long is C is non-empty, if x \in C then If is a non-empty subset, s is any scalar, and are any vectors then which implies: Examples The following counter-example demonstrates a continuous linear isomorphism A : H \to H for which Endow H := \R^2 with the dot product, let and for every real s \in \R, let be the line of slope s through the origin, where it is readily verified that Pick a real number r \neq 0 and define by (so this map scales the x-coordinate by r while leaving the y-coordinate unchanged). Then is an invertible continuous linear operator that satisfies and so that and Consequently, if C := L_s with s \neq 0 and if then