Normalized system

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)