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Normalized system

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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