Fréchet space
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- This article deals with Fréchet spaces in functional analysis. For Fréchet spaces in general topology, see T1 space. See also Fréchet-Urysohn space as a type of sequential space
In functional analysis and related areas of mathematics, Fréchet spaces or Frechet 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, in contrast, are locally convex spaces which are complete with respect to a translation invariant metric.
Fréchet spaces are studied because even though their topological structure is more complicated due to the lack of a norm, many important results in functional analysis, like the open mapping theorem and the Banach-Steinhaus theorem, still hold.
Spaces of infinitely often differentiable functions defined on compact sets are typical examples of Fréchet spaces.
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[edit] 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 space X is a Fréchet space if and only if it satisfies the following three properties:
- it is complete as a uniform space
- 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.
The alternative and somewhat more practical definition is the following: a topological vector space X is a Fréchet space if and only if it satisfies the following two properties:
- it is complete as a uniform space
- 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 (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.
[edit] Examples
Trivially, every Banach space is a Fréchet space as the norm induces a translation invariant metric and the space is complete with respect 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
- ||f||k = sup {|f (k)(x)| : x ∈ [0,1]}
for every integer k ≥ 0. Here, f (k) denotes the k-th 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 : 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 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.
Not all vector spaces with complete translation-variant metrics are Fréchet spaces. An example is Lp with p < 1. Of course, such spaces fail to be locally convex.
[edit] 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, the Gâteaux derivative:
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 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 Ck([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.
[edit] 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. Some of the relations between Lie algebras and Lie groups remain valid in this setting.
See the main article Fréchet manifold.
