View source for 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 <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]]).

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" />.

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" />,

<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>

in the metric of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070410/o07041022.png" />, 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/o07041023.png" /></td> </tr></table>

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

<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>

(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).

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]]]).

====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>



====Comments====


====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>