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A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677201.png" /> of elements of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677202.png" /> whose norms are all equal to one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677203.png" />. In particular, a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677204.png" /> of functions in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677205.png" /> is said to be normalized if
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<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/n/n067/n067720/n0677206.png" /></td> </tr></table>
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Normalization of a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677207.png" /> of non-zero elements of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677208.png" /> means the construction of a normalized system of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n0677209.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n06772010.png" /> are non-zero numbers, the so-called normalizing factors. As a sequence of normalizing factors one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067720/n06772011.png" />.
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A system  $  \{ x _ {i} \} $
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of elements of a [[Banach space|Banach space]]  $  B $
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whose norms are all equal to one,  $  \| x _ {i} \| _ {B} = 1 $.
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In particular, a system $  \{ f _ {i} \} $
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of functions in the space  $  L _ {2} [ a, b] $
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is said to be normalized if
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$$
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\int\limits _ { a } ^ { b }  | f _ {i} ( x) |  ^ {2}  dx  = 1.
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$$
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Normalization of a system  $  \{ x _ {i} \} $
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of non-zero elements of a Banach space $  B $
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means the construction of a normalized system of the form $  \{ \lambda _ {i} x _ {i} \} $,  
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where the $  \lambda _ {i} $
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are non-zero numbers, the so-called normalizing factors. As a sequence of normalizing factors one can take $  \lambda _ {i} = 1/ \| x _ {i} \| _ {B} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functionalanalysis in normierten Räumen" , Akademie Verlag  (1964)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


A system $ \{ x _ {i} \} $ of elements of a Banach space $ B $ whose norms are all equal to one, $ \| x _ {i} \| _ {B} = 1 $. In particular, a system $ \{ f _ {i} \} $ of functions in the space $ L _ {2} [ a, b] $ is said to be normalized if

$$ \int\limits _ { a } ^ { b } | f _ {i} ( x) | ^ {2} dx = 1. $$

Normalization of a system $ \{ x _ {i} \} $ of non-zero elements of a Banach space $ B $ means the construction of a normalized system of the form $ \{ \lambda _ {i} x _ {i} \} $, where the $ \lambda _ {i} $ are non-zero numbers, the so-called normalizing factors. As a sequence of normalizing factors one can take $ \lambda _ {i} = 1/ \| x _ {i} \| _ {B} $.

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[3] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian)
How to Cite This Entry:
Normalized system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normalized_system&oldid=48019
This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article