Mathematically correct, Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space *V* with a norm ||.|| such that every Cauchy sequence in *V* converges. See metric space for a discussion of the concept of completeness.

Every Hilbert space is a Banach space, but there are several other important examples. Consider for instance the space of all continuous functions *f* : [*a*, *b*] `->` **R** defined on a closed interval [*a*, *b*] with values in the real numbers **R**. The norm ||*f*|| of such a function can be defined as ||*f*|| = sup { |*f*(*x*)| : *x* in [*a*, *b*] }. This is indeed a norm since continuous functions defined on a closed interval are bounded. The space is complete under this norm, and the resulting Banach space is denoted by C[*a*, *b*]. Another commonly used Banach space is the space of all sequences (*a*_{n}) such that the series ∑ |*a*_{n}| converges; this sum is then defined to be the norm of the sequence. The space is complete under this norm and is denoted by *l*^{1}.

If *V* and *W* are Banach spaces, the set of all continuous linear maps *A* : *V* `->` *W* is denoted by L(*V*, *W*). This is a vector space, and by defining the norm ||*A*|| = sup { ||*Ax*|| : *x* in *V* with ||*x*|| ≤ 1 } it can be turned into a Banach space.

If *V* is a Banach space and **K** is the underlying field (either the real or the complex numbers), then **K** is itself a Banach space and we can define
the *dual space* *V'* by *V'* = L(*V*, **K**). This is again a Banach space. There is a natural map Φ from *V* to *V''* defined by

- Φ(
*x*)(φ) = φ(*x*)