Source : Free On-Line Dictionary of Computing

Banach space
     
         A {complete} {normed} {vector space}.  Metric is
        induced by the norm: d(x,y) = ||x-y||.  Completeness means
        that every {Cauchy sequence} converges to an element of the
        space.  All finite-dimensional {real} and {complex} normed
        vector spaces are complete and thus are Banach spaces.
     
        Using absolute value for the norm, the real numbers are a
        Banach space whereas the rationals are not.  This is because
        there are sequences of rationals that converges to
        irrationals.
     
        Several theorems hold only in Banach spaces, e.g. the {Banach
        inverse mapping theorem}.  All finite-dimensional real and
        complex vector spaces are Banach spaces.  {Hilbert spaces},
        spaces of {integrable functions}, and spaces of {absolutely
        convergent series} are examples of infinite-dimensional Banach
        spaces.  Applications include {wavelets}, {signal processing},
        and radar.
     
        [Robert E. Megginson, "An Introduction to Banach Space
        Theory", Graduate Texts in Mathematics, 183, Springer Verlag,
        September 1998].
     
        (2000-03-10)