The Free On-line Dictionary of Computing (30 December 2018):

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)