# Normalized system

From Encyclopedia of Mathematics

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 .

#### 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. A.A. Talalyan (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Normalized_system&oldid=18213

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098