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DEFINITION Banach spaces are defined as Complete Normed Vector Space s. This means that a Banach space is a Vector Space ''V'' over the Real or Complex numbers with a Norm ‖·‖ such that every Cauchy Sequence (with respect to the Metric ''d''(''x'', ''y'') = ‖''x'' − ''y''‖) in ''V'' has a Limit in ''V''. Since the norm induces a Topology on the vector space, a Banach space provides an example of a Topological Vector Space . EXAMPLES Throughout, let K stand for one of the Fields '''R''' or '''C'''. | ||
| Becomes A Banach Space If We Define The Norm Of Such A Function As ''f'' | sup { ''f''(''x'') : ''x'' in ''b'' } This is indeed a norm since continuous functions defined on a closed interval are bounded The space is complete under this norm, and the resulting Banach space is denoted by C ''b'' This example can be generalized to the space C(''X'') of all continuous functions ''X'' &rarr '''K''', where ''X'' is a Compact Space , or to the space of all ''bounded'' continuous functions ''X'' &rarr '''K''', where ''X'' is any Topological Space , or indeed to the space B(''X'') of all bounded functions ''X'' &rarr '''K''', where ''X'' is any Set In all these examples, we can multiply functions and stay in the same space: all these examples are in fact |
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| If ''p'' &ge 1 Is A Real Number, We Can Consider The Space Of All Infinite | "http://wwwinformationdelightinfo/encyclopedia/entry/sequence" class="copylinks">Sequence s (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ) of elements in '''K''' such that the Infinite Series &sum<sub>i</sub> ''x''<sub>''i''</sub><sup>''p''</sup> is finite The ''p''-th root of this series' value is then defined to be the ''p''-norm of the sequence The space, together with this norm, is a Banach space it is denoted by ''l<sup> p</sup>'' |
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| Again, If ''p'' &ge 1 Is A Real Number, We Can Consider All Functions ''f'' : | "''a''," class="copylinks" target="_blank">''b'' &rarr '''K''' such that ''f''<sup>''p''</sup> is es then forms a Banach space it is denoted by L<sup>'' p''</sup> ''b'' It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space These examples can be generalized see L<sup> ''p''</sup> Spaces for details |
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| Spaces, Not All Linear Maps Are Automatically Continuous L(''V'', ''W'') Is A Vector Space, And By Defining The Norm ''A'' | sup { ''Ax'' : ''x'' in ''V'' with ''x'' &le 1 } it can be turned into a Banach space |
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| As Mentioned Above, Every Hilbert Space Is A Banach Space Because, By Definition, A Hilbert Space Is Complete With Respect To The Norm Associated With Its Inner Product, Where A Norm And An Inner Product Are Said To Be Associated If '''v'''&2 | ('''v''','''v''') for all '''v''' |
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| :<math>\u+v\^2 + \u-v\^2 | 2(\u\^2 + \v\^2)</math> |
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| :<math>(u,v) | rac{1}{4} (\u+v\^2 - \u-v\^2)</math> |
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| :<math>(u,v) | rac{1}{4} \left(\u+v\^2 - \u-v\^2 + i(\u+iv\^2 - \u-iv\^2)
ight)</math> |
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