Orthonormality

In linear algebra and functional analysis, areas of mathematics, an orthogonal system is a set of vectors of a vector space with scalar product ( pre-Hilbert space ) that are pairwise perpendicular.

Definition

A subset of a Prähilbertraums is called orthogonal if:

So here denotes the scalar product of the space, the standard scalar product in Euclidean space.

Also are all vectors from normalized with standard, one speaks of a orthonormal system.

Properties

• Orthogonal systems are linearly independent.
• In separable Hilbert spaces ( in particular in all finite dimensional Hilbert spaces ) can be combined with the Gram- Schmidt orthogonalization of any linearly independent system an orthogonal (or orthonormal ) or from any ( shudder )-based construct an orthogonal (or orthonormal ) basis.
• For an orthonormal system Bessel's inequality holds
• For each vector the amount of for which holds a maximum countable.

Examples

• Im with the standard scalar product the standard basis is an orthogonal
• In the functions form an orthogonal system (see also trigonometric polynomial )
• In the scalar product, the sequences form an orthogonal system
• In the pre-Hilbert space of polynomials with degree less than or equal to 5, provided with the inner product, form the functions