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Orthonormal basis

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In mathematics, an '''orthonormal basis''' of an inner product space ''V'' (i.e., a vector space with an inner product), or in particular of a Hilbert space ''H'', is a set of elements whose linear span span is dense in the space, in which the elements are mutually orthogonality orthogonal and normal, that is, of magnitude 1. An '''orthogonal basis''' satisfies the same conditions, without the condition of length 1; it is easy to change the vectors in an '''''orthogonal''''' Basis (linear algebra) basis by scalar multiples to get an '''''orthonormal''''' basis, and indeed this is a typical way that an orthonormal basis is constructed, via an orthogonal basis. These concepts are important both for finite-dimensional and infinite-dimensional spaces. For finite-dimensional spaces the condition of a dense span is the same as 'span', as used in linear algebra. An orthonormal basis is ''not'' generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of ''finitely'' many members of an orthonormal basis. In the infinite-dimensional case the distinction matters: the definition given above requires only that the span of an orthonormal basis be ''dense in'' the vector space, not that it equal the entire space. An orthonormal basis of a vector space ''V'' makes no sense unless ''V'' is given an inner product; Banach spaces do not generally have orthonormal bases.

Examples
* The set {(1,0,0),(0,1,0),(0,0,1)} (the standard basis), as well as versions obtained by rotation about an axis through the origin or reflection in a plane through the origin, or a combination, each form an orthonormal basis of '''R'''³ * The set {''f''_''n'' : ''n'' ∈ '''Z'''} with ''f''_''n''(''x'') = exponential function exp(2π''inx'') forms an orthonormal basis of the complex space L²([0,1]). This is fundamental to the study of Fourier series. * The set {''e''_''b'' : ''b'' ∈ ''B''} with ''e''_''b''(''c'') = 1 if ''b''=''c'' and 0 otherwise forms an orthonormal basis of ''l''²(''B''). * Eigenfunctions of a Sturm-Liouville eigenproblem.

Basic formulae
If ''B'' is an orthogonal basis of ''H'', then every element ''x'' of ''H'' may be written as :

x=\sum_{b\in B}{\langle x,b\rangle\over||b||^2} b

When ''B'' is orthonormal, we have instead :

x=\sum_{b\in B}\langle x,b\rangle b

and the norm of ''x'' can be given by :

\|x\|^2=\sum_{b\in B}|\langle x,b\rangle |^2

. Even if ''B'' is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the ''Fourier expansion'' of ''x''. See also Generalized Fourier series. If ''B'' is an orthonormal basis of ''H'', then ''H'' is ''isomorphic'' to ''l''²(''B'') in the following sense: there exists a bijective linear operator linear map Φ : ''H'' -> ''l''²(''B'') such that :

\langle\Phi(x),\Phi(y)\rangle=\langle x,y\rangle

for all ''x'' and ''y'' in ''H''.

Incomplete orthogonal sets
Given a Hilbert space ''H'' and a set ''S'' of mutually orthogonal vectors in ''H'', we can take the smallest closed linear subspace ''V'' of ''H'' containing ''S''. Then ''S'' will be an orthogonal basis of ''V''; which may of course be smaller than ''H'' itself, being an ''incomplete'' orthogonal set, or be ''H'', when it is a ''complete'' orthogonal set.

Existence
Using Zorns lemma Zorn's lemma and the Gram-Schmidt process, one can show that ''every'' Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinal number cardinality. A Hilbert space is separable metric space separable if and only if it admits a countable orthonormal basis.

Relation to Hamel bases
Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases. (Hamel bases are of little practical interest in inner product spaces, while orthonormal bases are of major importance - the distinction may though shed light on what an orthonormal basis is.) See also Basis (linear algebra) and Schauder basis. Category: Functional analysis Category: Fourier analysis da:Ortonormal basis de:Orthonormalbasis it:Base ortonormale he:×ž×¢×¨×›×ª ×?×•×¨×ª×•× ×•×¨×ž×œ×™×ª ×©×œ×ž×” pl:Baza ortonormalna sv:Ortonormal bas

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