DefinitionA Banach space is a vector space together with a norm, in the topology of which the vector space is complete.

That is, there is a mapping from B to R+, denoted ||v||, such that

  1. ||v|| = || ||v||
  2. ||v + w|| ||v|| + ||w||
  3. ||v|| = 0 iff v = 0.

Further, all Cauchy sequences with respect to the norm converge to limits in B.

It is normal to assume tacitly that a Banach space is infinite dimensional.

Some Banach spaces are Hilbert spaces. They are called Hilbert spaces when the norm is defined by an inner product, <v, w> by ||v||2 := <v, v>. A norm can be defined in terms of an inner product iff it satisfies the parallelogram identity.