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Difference between revisions of "Orthogonality"

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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 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 $(
 
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} $
+
the countable sum  $  \sum_{i=1}  ^  \infty  x _ {i} $
 
is understood in the sense of convergence of the series in the metric of  $  H $),  
 
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} $(
+
then  $  \| x \|  ^ {2} = \sum_{i=1}  ^  \infty  \| x _ {i} \|  ^ {2} $(
 
see [[Parseval equality|Parseval equality]]).
 
see [[Parseval equality|Parseval equality]]).
  
 
A complete, countable, orthonormal system  $  \{ x _ {i} \} $
 
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 $
 
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} $,  
+
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} $
 
where  $  c _ {i} x _ {i} = ( x, x _ {i} ) x _ {i} $
 
is the orthogonal projection of the element  $  x $
 
is the orthogonal projection of the element  $  x $
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$$  
 
$$  
f  =  \sum _ { k= } 1 ^  \infty  c _ {k} \phi _ {k}  $$
+
f  =  \sum_{k=1} ^  \infty  c _ {k} \phi _ {k}  $$
  
 
in the metric of the space  $  L _ {2} [ a, b] $,  
 
in the metric of the space  $  L _ {2} [ a, b] $,  
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When the  $  \phi _ {k} $
 
When the  $  \phi _ {k} $
 
are bounded functions, the coefficients  $  c _ {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|Trigonometric system]]; [[Haar system|Haar system]]). With respect to functions, therefore, the term  "orthogonality"  is used in a broader sense: Two functions  $  f $
+
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|Haar system]]). With respect to functions, therefore, the term  "orthogonality"  is used in a broader sense: Two functions  $  f $
 
and  $  g $
 
and  $  g $
 
which are integrable on the segment  $  [ a, b] $
 
which are integrable on the segment  $  [ a, b] $

Latest revision as of 12:13, 6 January 2024


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

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)
How to Cite This Entry:
Orthogonality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonality&oldid=54866
This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article