Banach space
a vector sp-ce on which a norm is defined that is complete.
mathematics
a complete normed vector sp-ce. metric is induced by the norm: d(x,y) = ||x-y||. completeness means that every cauchy sequence converges to an element of the sp-ce. all finite-dimensional real and complex normed vector sp-ces are complete and thus are banach sp-ces.
using absolute value for the norm, the real numbers are a banach sp-ce whereas the rationals are not. this is because there are sequences of rationals that converges to irrationals.
several theorems hold only in banach sp-ces, e.g. the banach inverse mapping theorem. all finite-dimensional real and complex vector sp-ces are banach sp-ces. hilbert sp-ces, sp-ces of integrable functions, and sp-ces of absolutely convergent series are examples of infinite-dimensional banach sp-ces. applications include wavelets, signal processing, and radar.
[robert e. megginson, “an introduction to banach sp-ce theory”, graduate texts in mathematics, 183, springer verlag, september 1998].
(2000-03-10)
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