Projection set

This article explains the projection theorem of functional analysis; for other meanings, see projection sentence (disambiguation) .

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.

statement

Let be a closed subspace of a Hilbert space with the scalar product . Then there are all a precisely and exactly one with . ${\displaystyle {\mathcal {M}}\subset {\mathcal {H}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle f\in {\mathcal {H}}}$ ${\displaystyle f_{1}\in {\mathcal {M}}}$ ${\displaystyle f_{2}\in {\mathcal {M}}^{\perp }}$ ${\displaystyle f=f_{1}+f_{2}}$

It is for all the orthogonal complement of . The name projection set comes from the fact that the orthogonal projection is given by the assignment . ${\displaystyle {\mathcal {M}}^{\perp }:=\{g\in {\mathcal {H}}\mid \langle f,g\rangle =0}$ ${\displaystyle f\in {\mathcal {M}}\}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle f\mapsto f_{1}}$ ${\displaystyle {\mathcal {M}}}$

Evidence sketch

First you consider the distance to you . ${\displaystyle f\in {\mathcal {H}}\setminus {\mathcal {M}}}$ ${\displaystyle d:=\inf\{\|f-m\|:m\in {\mathcal {M}}\}}$ ${\displaystyle {\mathcal {M}}}$

There is a sequence with . With the help of the parallelogram equation one shows that is a Cauchy sequence . Since is closed and complete, converges to a with . ${\displaystyle m_{j}\in {\mathcal {M}}}$ ${\displaystyle \|f-m_{j}\|\rightarrow d}$ ${\displaystyle m_{j}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle H}$ ${\displaystyle m_{j}}$ ${\displaystyle m\in {\mathcal {M}}}$ ${\displaystyle \|f-m\|=d>0}$

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. ${\displaystyle f-m}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \langle f-m,m'\rangle =0}$ ${\displaystyle m'\in {\mathcal {M}}}$ ${\displaystyle f=m+(f-m)}$ ${\displaystyle {\mathcal {H}}={\mathcal {M}}+{\mathcal {M}}^{\perp }}$ ${\displaystyle g\in {\mathcal {M}}\cap {\mathcal {M}}^{\perp }}$ ${\displaystyle \langle g,g\rangle =0}$ ${\displaystyle g=0}$

Consequences

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 .

generalization

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 . ${\displaystyle C\subset {\mathcal {H}}}$ ${\displaystyle f\in {\mathcal {H}}}$ ${\displaystyle f_{1}\in C}$ ${\displaystyle \|f-f_{1}\|=\mathrm {min} \{\|f-g\|\colon g\in C\}}$

Individual evidence

1. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 11.7
2. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 11.4