| Orthonormal Basis |
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| CATEGORIES ABOUT ORTHONORMAL BASIS | |
| functional analysis | |
| fourier analysis | |
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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 Space s do not generally have orthonormal bases. EXAMPLES
BASIC FORMULAE If ''B'' is an orthogonal basis of ''H'', then every element ''x'' of ''H'' may be written as | ||
| :<math>\x\^2 | \sum_{b\in B}\langle x,b
angle ^2</math> |
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