Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

2.5 The projection theorem and orthogonal decomposition

It is safe to say that almost every statistical problem eventually leads to some type of optimization problem with the classical Gauss–Markov Theorem providing an important case in point. Optimization in vector and function spaces becomes much more tractable when there is an inherent geometry that can be exploited to aid in the characterization of extrema. This is undoubtedly why Hilbert spaces have occupied such a central role in statistics.

The following result is fundamental in optimization theory.

Theorem 2.5.1

Let be a closed convex set in a Hilbert space . For every , has a unique minimizer in that satisfies

2.13

for all .

 

Proof

Our proof follows along the lines of that in Luenberger (1969). First we observe that ...