In mathematics, the Hilbert projection theorem is a famous result of convex analysis in a Hilbert space

Finite dimensional case

Some intuition for the theorem can be obtained by considering the first order condition of the optimization problem.

is a or

In matrix derivative notation:

Statement

Detailed elementary proof

Proof by reduction to a special case

Consequences

:

:

:

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 following functions. Their notation will also be used to simplify certain statements.

A is equal to the

Effects of translations and scalings

Examples

with the dot product Pick a real number

Iterated projections

be the projection function.

. The Kaczmarz method can be approximated as an intersection of simple objects like half-spaces, hyperplanes, finite-dimensional subspaces, or cones.

are closed subspaces, then

See also

Notes

References

Bibliography