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)