A generalization of the concept of perpendicularity of vectors in a Euclidean space. The most natural concept of orthogonality is put forward in the theory of Hilbert spaces. Two elements and of a Hilbert space are said to be orthogonal if their inner product is equal to zero (). This concept of orthogonality in the particular case where is a Euclidean space coincides with the concept of perpendicularity of two vectors. In terms of this concept, in any Hilbert space Pythagoras' theorem holds: If an element is equal to a finite or countable sum of pairwise orthogonal elements (the countable sum is understood in the sense of convergence of the series in the metric of ), then (see Parseval equality).
A complete, countable, orthonormal system in a separable Hilbert space is the analogue of a complete system of pairwise orthonormal vectors in a finite-dimensional Euclidean space: Any element can be uniquely represented as the sum , where is the orthogonal projection of the element onto the span of the vector .
E.g., in the function space , if is a complete orthonormal system, then for every ,
in the metric of the space , where
When the are bounded functions, the coefficients can be defined as above for any integrable function. In these cases the question of the convergence of a corresponding series in one sense or another is of interest (see Trigonometric system; Haar system). With respect to functions, therefore, the term "orthogonality" is used in a broader sense: Two functions and which are integrable on the segment are orthogonal if
(for the integral to exist, it is usually required that , , , , where is the set of bounded measurable functions).
Definitions of orthogonality of elements of an arbitrary normed linear space also exist. One of them (see ) is as follows: An element of a real normed space is considered orthogonal to the element if for all real . In terms of this concept certain necessary and sufficient conditions have been established under which a scalar (inner) product of elements of can be defined (see , ).
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Orthogonality. A.A. Talalyan (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Orthogonality&oldid=11950