Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

* Banach fixed point theorem

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students ( for example in the Banach – Schauder theorem ) and other mathematicians ( in particular Bouwer and Poincaré and Birkhoff ).

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem.

Some examples are the Hahn – Banach theorem, König's lemma, Brouwer fixed point theorem, Gödel's completeness theorem and Jordan curve theorem.

The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed point theorem.

Unfortunately, the formula doesn't fulfill the preconditions for the Banach fixed point theorem, thus methods based on it don't work.

A general Banach – Mazur game is defined as follows: we have a topological space, a fixed subset, and a family of subsets of that satisfy the following properties.

An expression of prerequisites and proof of the existence of such solution is given by Banach fixed point theorem.

The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.

This version is essentially a special case of the Banach fixed point theorem.

The Schauder fixed-point theorem states, in one version, that if C is a nonempty closed convex subset of a Banach space V and f is a continuous map from C to C whose image is compact, then f has a fixed point.

* Banach fixed point theorem

* If Γ acts by isometries on a ( separable ) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E * invariant, then Γ has a fixed point in C.

It can then be shown, by using the Banach fixed point theorem, that the sequence of " Picard iterates " is convergent and that the limit is a solution to the problem.

shows that, thus proving the global uniqueness ( the local uniqueness is a consequence of the uniqueness of the Banach fixed point ).

We will proceed to apply Banach fixed point theorem using the metric on induced by the uniform norm

A key theme from the " categorical " point of view is that mathematics requires not only certain kinds of objects ( Lie groups, Banach spaces, etc.

The promised geometric property of reflexive Banach spaces is the following: if C is a closed non-empty convex subset of the reflexive space X, then for every x in X there exists a c in C such that || x − c || minimizes the distance between x and points of C. ( Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.

However, a result of Stefan Banach states that the set of functions which have a derivative at some point is a meager set in the space of all continuous functions.

However, the case of a compact operator on a Hilbert space ( or Banach space ) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0.

* A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon – Nikodym property, a closed and bounded set has an extreme point.

It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limit point.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

** The Banach – Alaoglu theorem about compactness of sets of functionals.

As a consequence of the Hahn – Banach theorem, this map is injective, and isometric.

The binomial theorem also holds for two commuting elements of a Banach algebra.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

* Hahn – Banach theorem

In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.

" Another version of Hahn – Banach theorem is known as Hahn – Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry.

It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier ( in 1912 ) by Eduard Helly, and a general extension theorem from which the Hahn – Banach theorem can be derived was proved in 1923 by Marcel Riesz.

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

* Open mapping theorem ( functional analysis ) or Banach – Schauder theorem states that a surjective continuous linear transformation of a Banach space X onto a Banach space Y is an open mapping

* A theorem of Gerald Edgar states that, in a Banach space with the Radon – Nikodym property, a closed and bounded set is the closed convex hull of its extreme points.

The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.

In functional analysis and related branches of mathematics, the Banach – Alaoglu theorem ( also known as Alaoglu's theorem ) states that the closed unit ball of the dual space of a normed vector space is compact in the weak * topology.

In mathematics, the Vitali – Hahn – Saks theorem, introduced by,, and, states that given μ < sub > n </ sub > for each integer n > 0, a countably additive function defined on a fixed sigma-algebra Σ, with values in a given Banach space B, such that