Orthogonal system

In linear algebra and functional analysis , sub-areas of mathematics , an orthogonal system is a set of vectors of a vector space with a scalar product ( prehilbert space ) that are perpendicular to each other in pairs. If the vectors are also normalized (i.e. they have norm 1), one speaks of an orthonormal system .

definition

A subset of a Prähilbert space is called an orthogonal system if: ${\displaystyle M}$ ${\displaystyle V}$

1. Two different vectors from are orthogonal to each other : ${\displaystyle M}$ ${\displaystyle \forall v,w\in M:v\neq w\Rightarrow \langle v,w\rangle =0}$
2. The set does not contain the zero vector .

Here the scalar product of the space denotes , in Euclidean space the standard scalar product . ${\displaystyle \langle v,w\rangle }$ ${\displaystyle V}$

Also applies

Each vector from is normalized, i.e. H. , ${\displaystyle M}$ ${\displaystyle \forall v\in M:\langle v,v\rangle =1}$

this is what is called an orthonormal system. ${\displaystyle M}$

properties

• Orthogonal systems are linearly independent .
• In separable Hilbert spaces (especially in all finite-dimensional Hilbert spaces), the Gram-Schmidt orthogonalization method can be used to construct an orthogonal system (or orthonormal system) from every linearly independent system or an orthogonal (or orthonormal) basis from every (Schauder) basis .
• For an orthonormal system , Bessel's inequality applies ${\displaystyle M}$

  ${\displaystyle \sum _{e\in M}|\langle x,\,e\rangle |^{2}\leq \|x\|^{2}\quad \forall ~x\in V.}$

• For each vector , the set for which applies is at most countable. ${\displaystyle x\in V}$ ${\displaystyle e\in M}$ ${\displaystyle \langle x,e\rangle \neq 0}$

Examples

• In using the standard scalar product, the standard basis is an orthogonal system ${\displaystyle \mathbb {R} ^{n}}$
• In the functions form an orthogonal system (see also trigonometric polynomial ) ${\displaystyle L^{2}([0,2\pi ])}$ ${\displaystyle \cos(kx)}$
• In the dot product of the sequences form an orthogonal system ${\displaystyle \ell ^{2}}$ ${\displaystyle (a,b)\mapsto \sum a_{n}b_{n}}$ ${\displaystyle (0,\cdots ,0,1,0,\cdots )}$
• In the Prähilbert space of polynomials with degrees less than or equal to 5 , provided with the -Scalar product , form the functions ${\displaystyle {\mathcal {P}}^{5}([0,1])}$ ${\displaystyle L^{2}}$ ${\displaystyle (a,b)\mapsto \int _{0}^{1}ab}$

${\displaystyle x\mapsto 1}$ and ${\displaystyle x\mapsto x-{\frac {1}{2}}}$

an orthogonal system.

See also

• Complete orthonormal system

literature

• Dirk Werner : Functional Analysis . 6th corrected edition. Springer, Berlin 2007, ISBN 978-3-540-72533-6 . Chapter V.3 (For the infinite-dimensional case, there is also evidence for the examples)
• Gerd Fischer: Linear Algebra: An Introduction for New Students . 13th edition. Vieweg, 2002, ISBN 3-528-97217-3 . (For the finite-dimensional case, there under "generating system")