Orthogonality

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 [4]) 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 [5], [6]).

References

[1]	L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian)
[2]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988)
[3]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[4]	G. Birkhoff, "Orthogonality in linear metric spaces" Duke Math. J. , 1 (1935) pp. 169–172
[5]	R. James, "Orthogonality and linear functionals in normed linear spaces" Trans. Amer. Math. Soc. , 61 (1947) pp. 265–292
[6]	R. James, "Inner products in normed linear spaces" Bull. Amer. Math. Soc. , 53 (1947) pp. 559–566

References

[a1]	D. Amir, "Characterizations of inner product spaces" , Birkhäuser (1986)
[a2]	N.I. [N.I. Akhiezer] Achieser, I.M. [I.M. Glaz'man] Glasman, "Theorie der linearen Operatoren im Hilbert Raum" , Akademie Verlag (1958) (Translated from Russian)
[a3]	V.I. Istrăţescu, "Inner product structures" , Reidel (1987)