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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 Span is Dense in the space, in which the elements are mutually 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 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 . For an (infinite-dimensional) Hilbert space, an orthonormal basis is ''not'' a Hamel Basis (a basis in the sense of linear algebra), i.e., it is not 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. For a finite-dimensional space, an orthonormal basis is a Hamel Basis . An orthonormal basis of a vector space ''V'' makes no sense unless ''V'' is given an inner product; a Banach Space does not have an orthonormal basis unless it is a Hilbert space. EXAMPLES
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:<math>\x\^2 |
\sum_{b\in B}\langle x,b
angle ^2</math> |
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