Banach space
<mathematics> A complete normed vector space. Metric is induced by the
norm: d(x,y) = xy. Completeness means that every Cauchy sequence converges
to an element of the space. All finitedimensional 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 finitedimensional real and complex vector spaces are Banach
spaces. Hilbert spaces, spaces of integrable functions, and spaces of absolutely
convergent series are examples of infinitedimensional 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].
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Nearby terms:
bamf « Banach algebra « Banach inverse mapping
theorem «
Banach space » BanachTarski paradox » banana
label » banana problem
