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Frchet SpaceThis article deals with Frchet spaces in functional analysis. For Frchet spaces in general topology, see T1 space.
In functional analysis, Frchet spaces are certain topological vector spaces more general than, but with some similarities to, Banach spaces. Spaces of infinitely often differentiable functions defined on compact sets are typical examples. Frchet spaces are named after the French mathematician Maurice Frchet. Definitions Frchet 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 Frchet 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 × 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 Frchet 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 Frchet 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 (xn) in X converges to x in the Frchet 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 Frchet 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 Frchet 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 Frchet 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 Frchet 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 Frchet 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 Frchet spaces; examples are the closed graph theorem and the open mapping theorem. If X and Y are Frchet spaces, then the space L(X,Y) consisting of all continuous linear maps from X to Y is not a Frchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Frchet spaces and necessitates a different definition for continuous differentiability of functions defined on Frchet spaces: Suppose X and Y are Frchet 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 Frchet spaces is again a Frchet 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 Frchet 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 Frchet spaces; a partial substitute is the Nash-Moser theorem. Frchet manifolds and Lie groups One may define Frchet manifolds as spaces that "locally look like" Frchet 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.
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