Orthogonal system

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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 .


A subset of a Prähilbert space is called an orthogonal system if:

  1. Two different vectors from are orthogonal to each other :
  2. The set does not contain the zero vector .

Here the scalar product of the space denotes , in Euclidean space the standard scalar product .

Also applies

Each vector from is normalized, i.e. H. ,

this is what is called an orthonormal system.


  • 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
  • For each vector , the set for which applies is at most countable.


  • In using the standard scalar product, the standard basis is an orthogonal system
  • In the functions form an orthogonal system (see also trigonometric polynomial )
  • In the dot product of the sequences form an orthogonal system
  • In the Prähilbert space of polynomials with degrees less than or equal to 5 , provided with the -Scalar product , form the functions
an orthogonal system.

See also


  • 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")