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Banach space

induced by the norm: d(x,y) = ||x-y||. Completeness means

that every Cauchy sequence converges to an element of the

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 Banachinverse mapping theorem. All finite-dimensional real and

complex vector spaces are Banach spaces. Hilbert spaces,

spaces of integrable functions, and spaces of absolutelyconvergent 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].