Difference between revisions of "Normalized system"
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| − | + | A system $ \{ x _ {i} \} $ | |
| + | of elements of a [[Banach space|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==== | ====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
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