# 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.