In mathematics, **Banach spaces**, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions.

Table of contents |

2 Examples 3 Linear operators 4 Derivatives 5 Dual space 6 Generalizations 7 External links |

### Definition

Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space *V* over the real or complex numbers with a norm ||.|| such that every Cauchy sequence (with respect to the metric *d*(*x*, *y*) = ||*x* - *y*||) in *V* has a limit in *V*.

### Examples

Throughout, let **K** stand for one of the fields **R** or **C**.

The familiar Euclidean spaces **K**^{n}, where the Euclidean norm of *x* = (*x*_{1}, ..., *x*_{n}) is given by ||*x*|| = (∑ |*x*_{i}|^{2})^{1/2}, is a Banach space.

The space of all continuous functions *f* : [*a*, *b*]
`->` **K** defined on a closed interval [*a*, *b*]
becomes a Banach space if we define the norm of such a function 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*]. This example can be
generalized to the space C(*X*) of all continuous functions *X*
`->` **K**, where *X* is a compact space, or to the
space of all *bounded* continuous functions *X* `->`
**K**, where *X* is any topological space, or indeed to the
space B(*X*) of all bounded functions *X* `->` **K**,
where *X* is any set. In all these examples, we can multiply
functions and stay in the same space: all these examples are in fact
unitary Banach algebras.

If *p* ≥ 1 is a real number, we can consider the space of all infinite sequences (*x*_{1}, *x*_{2}, *x*_{3}, ...) of elements in **K** such that the infinite series ∑ |*x*_{i}|^{p} converges. The *p*-th root of this series' value is then defined to be the *p*-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by *l ^{ p}*.

The Banach space *l ^{∞}* consists of all bounded sequences of elements in

**K**; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members.

Again, if *p* ≥ 1 is a real number, we can consider all functions *f* : [*a*, *b*] `->` **K** such that |*f*|^{p} is Lebesgue integrable. The *p*-th root of this integral is then defined to be the norm of *f*. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: *f* and *g* are equivalent if and only if the norm of *f* - *g* is zero. The set of equivalence classes then forms a Banach space; it is denoted by L^{ p}[*a*, *b*]. It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see L^{ p} spaces for details.

Finally, every Hilbert space is a Banach space. The converse is not true.

### Linear operators

If *V* and *W* are Banach spaces over the same ground field **K**, the set of all continuous
**K**-linear maps *A* : *V* `->` *W*
is denoted by L(*V*, *W*). Note that in infinite-dimensional
spaces, not all linear maps are automatically continuous. L(*V*, *W*) 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.

The space L(*V*) = L(*V*, *V*) even forms a unitary Banach algebra; the multiplication operation is given by the composition of linear maps.

### Derivatives

It is possible to define the derivative of a function *f* : *V* `->` *W* between two Banach spaces. Intuitively, if *x* is an element of *V*, the derivative of *f* at the point *x* is a continuous linear map which approximates *f* near *x*.

Formally, *f* is called *differentiable* at *x* if there exists a continuous linear map *A* : *V* `->` *W* such that

- lim
_{h->0}||*f*(*x*+*h*) -*f*(*x*) -*A*(*h*)|| / ||*h*|| = 0

*V*which converge to 0. If the limit exists, we write D

*f*(

*x*) =

*A*and call it the derivative of

*f*at

*x*.

This notion of derivative is in fact a generalization of the ordinary derivative of functions **R** `->` **R**, since the linear maps from **R** to **R** are just multiplications with real numbers.

If *f* is differentiable at *every* point *x* of *V*, then D*f* : *V* `->` L(*V*, *W*) is another map between Banach spaces (in general *not* a linear map!), and can possibly be differentiated again, thus defining the higher derivatives of *f*. The *n*-th derivative at a point *x* can then be viewed as a multilinear map *V ^{n}*

`->`

*W*.

Differentiation is a linear operation in the following sense: if *f* and *g* are two maps *V* `-` *W* which are differentiable at *x*, and *r* and *s* are scalars from **K**, then *rf* + *sg* is differentiable at *x* with D(*rf + sg*)(*x*) = *r*D(*f*)(*x*) + *s*D(*g*)(*x*).

The chain rule is also valid in this context: if *f* : *V* `->` *W* is differentiable at *x* in *V*, and *g* : *W* `->` *X* is differentiable in *f*(*x*), then the composition *g* o *f* is differentiable in *x* and the derivative is the composition of the derivatives:

- D(
*g*o*f*)(*x*) = D(*g*)(*f*(*x*)) o D(*f*)(*x*)

### Dual 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 (using the absolute value as norm) and we can define
the *dual space* *V'* by *V'* = L(*V*, **K**). This is again a Banach space. It can be used to define a new topology on *V*: the weak topology.

There is a natural map *F* from *V* to *V''* defined by

*F*(*x*)(*f*) =*f*(*x*)

*x*in

*V*and

*f*in

*V'*. As a consequence of the Hahn-Banach theorem, this map is injective; if it is also surjective, then the Banach space

*V*is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

### Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions **R** `->` **R** or the space of all distributions on **R**, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.