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 $ x $ and $ y $ of a Hilbert space $ H $ are said to be orthogonal $ ( x \perp y) $ if their inner product is equal to zero ( $ ( x, y) = 0 $). This concept of orthogonality in the particular case where $ H $ 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 $ x \in H $ is equal to a finite or countable sum of pairwise orthogonal elements $ x _ {i} \in H $( the countable sum $ \sum_{i=1} ^ \infty x _ {i} $ is understood in the sense of convergence of the series in the metric of $ H $), then $ \| x \| ^ {2} = \sum_{i=1} ^ \infty \| x _ {i} \| ^ {2} $( see Parseval equality).

A complete, countable, orthonormal system $ \{ x _ {i} \} $ in a separable Hilbert space is the analogue of a complete system of pairwise orthonormal vectors in a finite-dimensional Euclidean space: Any element $ x \in H $ can be uniquely represented as the sum $ \sum_{i=1} ^ \infty c _ {i} x _ {i} $, where $ c _ {i} x _ {i} = ( x, x _ {i} ) x _ {i} $ is the orthogonal projection of the element $ x $ onto the span of the vector $ x _ {i} $.

E.g., in the function space $ L _ {2} [ a, b] $, if $ \{ \phi _ {k} \} $ is a complete orthonormal system, then for every $ f \in L _ {2} [ a, b] $,

$$ f = \sum_{k=1} ^ \infty c _ {k} \phi _ {k} $$

in the metric of the space $ L _ {2} [ a, b] $, where

$$ c _ {k} = \int\limits _ { a } ^ { b } f ( x) \overline{ {\phi _ {k} ( x) }}\; dx. $$

When the $ \phi _ {k} $ are bounded functions, the coefficients $ c _ {k} $ 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 $ f $ and $ g $ which are integrable on the segment $ [ a, b] $ are orthogonal if

$$ \int\limits _ { a } ^ { b } f( x) g( x) dx = 0 $$

(for the integral to exist, it is usually required that $ f \in L _ {p} [ a, b] $, $ 1 \leq p \leq \infty $, $ g \in L _ {q} [ a, b] $, $ p ^ {- 1 } + q ^ {- 1 } = 1 $, where $ L _ \infty [ a, b] $ 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 $ x $ of a real normed space $ B $ is considered orthogonal to the element $ y $ if $ \| x \| \leq \| x + ky \| $ for all real $ k $. In terms of this concept certain necessary and sufficient conditions have been established under which a scalar (inner) product of elements of $ B $ 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)