Fréchet Space Definitions Examples Properties And Further

Fréchet spaces can be defined in two equivalent ways. The first employs a translation-invariant metric, the second a countable family of semi-norms.

A topological vector X space is a Fréchet space iff it satisfies the following three properties:

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.

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:

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.

2 Examples

The vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for C^∞([0,1]) of all infinitely often differentiable functions f : [0,1] → R becomes a Fréchet space with the seminorms

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 convergesIn 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 sequence's elements, as their index approaches infinity. Limits are used in calculu towards the element f of C^∞([0,1]) if and only if for every integer k≥0, the sequence (f_n^(k)) converges uniformlyCalculus In mathematical analysis, a sequence { f } of functions converges uniformly to a limiting function f if the speed of convergence of f ''x to f ''x does not depend on x''. This notion is used because several important properties of the functions f towards f^(k).

More generally, if M is a compact C^∞ manifoldIn mathematics, a manifold ''M is a type of space, characterised 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. Therefor 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.

The space of all sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, ors of real numbers becomes a Fréchet space if we define the k-th semi-norm of a sequence to be the absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno of the k-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence.

3 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.

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:

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

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

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

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 .

4 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.

Functional analysis

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Home > Fréchet space

1 Definitions

2 Examples

3 Properties and further notions

4 Fréchet manifolds and Lie groups

Topics: Fréchet Space, Definitions, Examples, Properties And Further...