This article deals with Fréchet spaces in functional analysis. For Fréchet spaces in general topology, see T1 space. In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... The title given to this article is incorrect due to technical limitations. ...

In functional analysis and related areas of mathematics Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces, normed vector spaces which are complete with respect to the metric induced by the norm. Fréchet spaces are locally convex spaces which are complete with respect to a translation invariant metric. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Maurice Fréchet (born September 2, 1878, died June 4, 1973) was a French mathematician. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... In mathematics, given a vector space V over K, a norm on V is a function ||·||:V->R; x->||x|| with the following properties: For all a ∈ K and all u and v ∈ V, 1. ...

Fréchet space are studied because even though their topological structure is more complicated due to the translation invariant metric, many important results in functional analysis, like the open mapping theorem and the Banach-Steinhaus theorem, still hold. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, there are two theorems with the name open mapping theorem. Functional analysis In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. ...

Spaces of infinitely often differentiable functions defined on compact sets are typical examples of Fréchet spaces. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...

Definitions

Fréchet spaces can be defined in two equivalent ways. The first employs a translation-invariant metric, the second a countable family of semi-norms. In mathematics the term countable set is used to describe the size of a set, e. ... In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...

A topological vector X space is a Fréchet space iff it satisfies the following three properties: ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...

• it is complete
• it is locally convex
• its topology can be induced by a translation invariant metric, i.e. a metric d : X × X → R such that d(x,y) = d(x+a, y+a) for all a,x,y in X. This means that a subset U of X is open if and only if for every u in U there exists an ε>0 such that {v : d(u,v) < ε} is a subset of U.

Note that there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

The alternative and somewhat more practical definition is the following: a topological vector X space is a Fréchet space iff it satisfies the following two properties:

• it is complete
• its topology may be induced by a countable family of semi-norms ||.||_k, k = 0,1,2,... This means that a subset U of X is open if and only if for every u in U there exists K≥0 and ε>0 such that {v : ||u - v||_k < ε for all k ≤ K} is a subset of U.

A sequence (x_n) in X converges to x in the Fréchet space defined by a family of semi-norms if and only if it converges to x with respect to each of the given semi-norms.

Examples

Trivially every Banach space is a Fréchet space as a norm induces a translation invariant metric and a Banach space is complete with repects to this metric.

The vector space C^∞([0,1]) of all infinitely often differentiable functions f : [0,1] → R becomes a Fréchet space with the seminorms The fundamental concept in linear algebra is that of a vector space or linear space. ...

||f||_k = sup {|f ^(k)(x)| : x ∈ [0,1]}

for every integer k ≥ 0. Here, f ^(k) denotes the k-the derivative of f, and f ⁽⁰⁾ = f. In this Fréchet space, a sequence (f_n) of functions converges towards the element f of C^∞([0,1]) if and only if for every integer k≥0, the sequence (f_n^(k)) converges uniformly towards f ^(k). In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ...

More generally, if M is a compact C^∞ manifold and B is a Banach space, then the set of all infinitely often differentiable functions f : M → B can be turned into a Fréchet space; the seminorms are given by the suprema of the norms of all partial derivatives. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

The space ω of real valued sequences becomes a Fréchet space if we define the k-th semi-norm of a sequence to be the absolute value of the k-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence. Graph of absolute value function In mathematics, the absolute value (or modulus) of a number is the difference between that number and 0. ...

Properties and further notions

Several important tools of functional analysis which are based on the Baire category theorem remain true in Fréchet spaces; examples are the closed graph theorem and the open mapping theorem. In mathematics, the Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. ... In mathematics, the closed graph theorem is a basic result of functional analysis. ... In mathematics, there are two theorems with the name open mapping theorem. Functional analysis In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y...

If X and Y are Fréchet spaces, then the space L(X,Y) consisting of all continuous linear maps from X to Y is not a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces: In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

Suppose X and Y are Fréchet spaces, U is an open subset of X, P : U → Y is a function, x∈U and h∈X. We say that P is differentiable at x in the direction h if the limit In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...

exists. We call P continuously differentiable in U if

is continuous. Since the product of Fréchet spaces is again a Fréchet space, we can then try to differentiate D(P) and define the higher derivatives of P in this fashion.

The derivative operator P : C^∞([0,1]) → C^∞([0,1]) defined by P(f) = f ' is itself infinitely often differentiable. The first derivative is given by

D(P)(f)(h) = h'

for any two elements f and h in C^∞([0,1]). This is a major advantage of the Fréchet space C^∞([0,1]) over the Banach space C^k([0,1]) for finite k.

If P : U → Y is a continuously differentiable function, then the differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

need not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces.

The inverse function theorem is not true in Fréchet spaces; a partial substitute is the Nash-Moser theorem. In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

Fréchet manifolds and Lie groups

One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like Euclidean space Rⁿ), and one can then extend the concept of Lie group to these manifolds. This is useful because for a given (ordinary) compact C^∞ manifold M, the set of all C^∞ diffeomorphisms f : M → M forms a generalized Lie group in this sense, and this Lie group captures the symmetries of M. The relation between Lie algebra and Lie group remains valid in this setting. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...