Hilbert projection theorem

In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector

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

that represents an arbitrary tangent direction, it follows that

be two integers, then the following equalities are true:

of a median in a triangle with sides of length

By giving an upper bound to the first two terms of the equality and by noticing that the middle of

Proof that the condition is sufficient: Let

It suffices to prove the theorem in the case of

because the general case follows from the statement below by replacing

Hilbert projection theorem (case

implies that the right hand side (RHS) of the above inequality can be made arbitrary close to

[note 2] The same must consequently also be true of the inequality's left hand side

The existence of the sequence follows from the definition of the infimum, as is now shown.

is a non-empty subset of non-negative real numbers and

(This first part of the proof works for any non-empty subset of

Lemma 1 can be used to prove uniqueness as follows.

of odd indices is the constant sequence

are unique and this constant subsequence also converges to

is a closed vector subspace of a Hilbert space

which is continuous and linear because this is true of each of its coordinates

The kernel of any linear map is a vector subspace of its domain, which is why

The Hilbert projection theorem guarantees the existence of a unique

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.

Effects of translations and scalings When this global minimum point

The Hilbert projection theorem guarantees that this unique minimum point exists whenever

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

Examples The following counter-example demonstrates a continuous linear isomorphism

is an invertible continuous linear operator that satisfies