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Fréchet space

In functional analysis, Fréchet spaces are certain topological vector spaces more general than, but with some similarities to, Banach spaces. Spaces of inifinitely often differentiable functions defined on compact sets are typical examples.

(Note that there is a completely unrelated concept in general topology, also named after the French mathematician Maurice Fréchet[?]: a topological space is called "Fréchet" or "accessible" if it satisfies the T1 separation axiom.)

Table of contents

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.

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

  • it is complete
  • it is locally convex[?]
  • its topology can be induced by a translation invariant metric, i.e. a metric d : X × XR 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.

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 kK} is a subset of U.

A sequence (xn) 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

The vector space C([0,1]) of all infinitely often differentiable functions f : [0,1] → R becomes a Fréchet space with the seminorms

||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 (fn) of functions converges towards the element f of C([0,1]) if and only if for every integer k≥0, the sequence (fn(k)) converges uniformly towards f (k).

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 : MB 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 sequences of real numbers 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.

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 : UY is a function, xU and hX. We say that P is differentiable at x in the direction h if the limit

<math>D(P)(x)(h) = \lim_{t\to 0} \,\frac{1}{t}\Big(P(x+th)-P(x)\Big)</math>
exists. We call P continuously differentiable in U if
<math>D(P):U\times X \to Y</math>
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

<math>D(P)(f)(h) = h'</math>
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 Ck([0,1]) for finite k.

If P : UY is a continuously differentiable function, then the differential equation

<math>x'(t) = P(x(t)),\quad x(0) = x_0\in U</math>
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[?].

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 : MM 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.



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