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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704102.png" /> of a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704103.png" /> are said to be orthogonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704104.png" /> if their [[Inner product|inner product]] is equal to zero (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704105.png" />). This concept of orthogonality in the particular case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704106.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704107.png" /> is equal to a finite or countable sum of pairwise orthogonal elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704108.png" /> (the countable sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o0704109.png" /> is understood in the sense of convergence of the series in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041010.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041011.png" /> (see [[Parseval equality|Parseval equality]]).
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A complete, countable, orthonormal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041012.png" /> in a separable Hilbert space is the analogue of a complete system of pairwise orthonormal vectors in a finite-dimensional Euclidean space: Any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041013.png" /> can be uniquely represented as the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041015.png" /> is the orthogonal projection of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041016.png" /> onto the span of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041017.png" />.
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{{TEX|auto}}
 +
{{TEX|done}}
  
E.g., in the function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041018.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041019.png" /> is a complete orthonormal system, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041020.png" />,
+
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|Hilbert space]]  $  H $
 +
are said to be orthogonal  $  ( x \perp  y) $
 +
if their [[Inner product|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|Parseval equality]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041021.png" /></td> </tr></table>
+
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} $.
  
in the metric of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041022.png" />, where
+
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] $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041023.png" /></td> </tr></table>
+
$$
 +
= \sum _ { k= } 1 ^  \infty  c _ {k} \phi _ {k}  $$
  
When the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041024.png" /> are bounded functions, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041025.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041027.png" /> which are integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041028.png" /> are orthogonal if
+
in the metric of the space  $  L _ {2} [ a, b] $,  
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041029.png" /></td> </tr></table>
+
$$
 +
c _ {k}  = \int\limits _ { a } ^ { b }  f ( x) \overline{ {\phi _ {k} ( x) }}\; dx.
 +
$$
  
(for the integral to exist, it is usually required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041034.png" /> is the set of bounded measurable functions).
+
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|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 $
 +
which are integrable on the segment  $  [ a, b] $
 +
are orthogonal if
  
Definitions of orthogonality of elements of an arbitrary normed linear space also exist. One of them (see [[#References|[4]]]) is as follows: An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041035.png" /> of a real normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041036.png" /> is considered orthogonal to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041038.png" /> for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041039.png" />. In terms of this concept certain necessary and sufficient conditions have been established under which a scalar (inner) product of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041040.png" /> can be defined (see [[#References|[5]]], [[#References|[6]]]).
+
$$
 +
\int\limits _ { a } ^ { b }  f( x) g( x)  dx  =  0
 +
$$
 +
 
 +
(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 [[#References|[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 [[#References|[5]]], [[#References|[6]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Birkhoff,  "Orthogonality in linear metric spaces"  ''Duke Math. J.'' , '''1'''  (1935)  pp. 169–172</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. James,  "Orthogonality and linear functionals in normed linear spaces"  ''Trans. Amer. Math. Soc.'' , '''61'''  (1947)  pp. 265–292</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. James,  "Inner products in normed linear spaces"  ''Bull. Amer. Math. Soc.'' , '''53'''  (1947)  pp. 559–566</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Birkhoff,  "Orthogonality in linear metric spaces"  ''Duke Math. J.'' , '''1'''  (1935)  pp. 169–172</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. James,  "Orthogonality and linear functionals in normed linear spaces"  ''Trans. Amer. Math. Soc.'' , '''61'''  (1947)  pp. 265–292</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. James,  "Inner products in normed linear spaces"  ''Bull. Amer. Math. Soc.'' , '''53'''  (1947)  pp. 559–566</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Amir,  "Characterizations of inner product spaces" , Birkhäuser  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Istrăţescu,  "Inner product structures" , Reidel  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Amir,  "Characterizations of inner product spaces" , Birkhäuser  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Istrăţescu,  "Inner product structures" , Reidel  (1987)</TD></TR></table>

Revision as of 08:04, 6 June 2020


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=11950
This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article