**Definition**A *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

- ||
**v**|| = || ||**v**|| - ||
**v**+**w**|| ||**v**|| + ||**w**|| - ||
**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.