The projection theorem is one of the most important theorems in functional analysis . Ultimately, it is used to constructively solve partial differential equations . It is an example of how geometric considerations lead to particularly far-reaching results in functional analysis . Ultimately, a vector is broken down into two components with respect to a given subspace . One component lies in the given subspace and the other is perpendicular to it. The first component is said to be the orthogonal projection of the vector onto the subspace.
First you consider the distance to you .
Now you show that it is perpendicular to , i.e. that applies to all . With you get . As for true , ie , the sum is direct.
Note that the proof only makes use of the Hilbert space axioms and is elementary in this regard, albeit very abstract. Thus the projection theorem applies in every Hilbert space. In addition to the consequences mentioned above, the functioning of the Gram-Schmidt orthonormalization method is ensured by this theorem . The projection theorem leads to the existence of a complete orthonormal system in Hilbert spaces. Finally, the projection theorem is one of the most important tools in proving the Fréchet-Riesz representation theorem .
Let be a closed, convex , non-empty subset of a Hilbert space. Then there is exactly one for each , so that the distance is minimal, so it applies .