Normalized system

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A system of elements of a Banach space whose norms are all equal to one, . In particular, a system of functions in the space is said to be normalized if

Normalization of a system of non-zero elements of a Banach space means the construction of a normalized system of the form , where the are non-zero numbers, the so-called normalizing factors. As a sequence of normalizing factors one can take .


[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. A.A. Talalyan (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098