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
,
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in the metric of the space , where
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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
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(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 |
Comments
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) |
Orthogonality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonality&oldid=11950