Hilbert projection theorem


In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector in a Hilbert space and every nonempty closed convex, there exists a unique vector for which is minimized over the vectors.
This is, in particular, true for any closed subspace of. In that case, a necessary and sufficient condition for is that the vector be orthogonal to.

Proof

Let δ be the distance between x and C, a sequence in C such that the distance squared between x and yn is below or equal to δ2 + 1/n. Let n and m be two integers, then the following equalities are true:
and
We have therefore:
By giving an upper bound to the first two terms of the equality and by noticing that the middle of yn and ym belong to C and has therefore a distance greater than or equal to δ from x, one gets :
The last inequality proves that is a Cauchy sequence. Since C is complete, the sequence is therefore convergent to a point y in C, whose distance from x is minimal.
Let y1 and y2 be two minimizers. Then:
Since belongs to C, we have and therefore
Hence, which proves uniqueness.
The condition is sufficient:
Let such that for all.
which proves that is a minimizer.
The condition is necessary:
Let be the minimizer. Let and.
is always non-negative. Therefore,
QED