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 spaceX is a Fréchet spaceif and only if it satisfies the following three properties:
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 = d 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 is a subset of U.
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 three properties:
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 is a subset of U.
A family of seminorms on yields a Hausdorff topology if and only if A sequence 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.
Constructing Fréchet spaces
Recall that a seminorm ǁ ⋅ ǁ is a function from a vector space X to the real numbers satisfying three properties. For all x and y in X and all scalars c, If ǁxǁ = 0 actually implies that x = 0, then ǁ ⋅ ǁ is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows: To construct a Fréchet space, one typically starts with a vector space X and defines a countable family of semi-norms ǁ ⋅ ǁk on X with the following two properties:
if x ∈ X and ǁxǁk = 0 for all k ≥ 0, then x = 0;
if is a sequence in X which is Cauchy with respect to each semi-norm ǁ ⋅ ǁk, then there exists x ∈ X such that converges to x with respect to each semi-norm ǁ ⋅ ǁk.
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∞ of all infinitely differentiable functions ƒ: → R becomes a Fréchet space with the seminorms
The vector space C∞ of all infinitely differentiable functions ƒ: R → R becomes a Fréchet space with the seminorms
The vector space Cm of all m-times continuously differentiable functions ƒ: R → R becomes a Fréchet space with the seminorms
Let H be the space of entire functions on the complex plane. Then the family of seminorms
Let H be the space of entire functions of exponential type τ. Then the family of seminorms
If M is a compactC∞-manifold and B is a Banach space, then the set C∞ of all infinitely-often differentiable functions ƒ: M → B can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If M is a C∞-manifold which admits a countable sequence Kn of compact subsets, so that every compact subset of M is contained in at least one Kn, then the spaces Cm and C∞ are also Fréchet space in a natural manner.
The spaceRω of all 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-invariant metrics are Fréchet spaces. An example is the Lp space|space Lp with p < 1. This space fails to be locally convex. It is an F-space.
Properties and further notions
If a Fréchet space admits a continuous norm, we can take all the seminorms to be norms by adding the continuous norm to each of them. A Banach space, C∞, C∞ with X compact, and H all admit norms, while Rω and C do not. A closed subspace of a Fréchet space is a Fréchet space. A quotient of a Fréchet space by a closed subspace is a Fréchet space. The direct sum of a finite number of Fréchet spaces is a Fréchet space. 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. All Fréchet spaces are stereotype. In the theory of stereotype spaces Fréchet spaces are dual objects to Brauner spaces. Every bounded linear operator from a Fréchet space into another topological vector space is continuous. There exists a Fréchet space having a bounded subset and also a dense vector subspace such that is not contained in the closure of any bounded subset of.
Differentiation of functions
If X and Y are Fréchet spaces, then the space L 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 Gateaux 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 Pcontinuously differentiable in U if is continuous. Since the product of Fréchet spaces is again a Fréchet space, we can then try to differentiate D and define the higher derivatives of P in this fashion. The derivative operator P : C∞ → C∞ defined by P = ƒ′ is itself infinitely differentiable. The first derivative is given by for any two elements ƒ and h in C∞. This is a major advantage of the Fréchet space C∞ over the Banach space Ck 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.
Fréchet manifolds and Lie groups
One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces, and one can then extend the concept of Lie group to these manifolds. This is useful because for a given compact C∞ manifold M, the set of all C∞ diffeomorphisms ƒ: 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. Another important example of a Fréchet Lie group is the loop group of a compact Lie group G, the smooth mappings γ : S1 → G, multiplied pointwise by = γ1 γ2.
Generalizations
If we drop the requirement for the space to be locally convex, we obtain F-spaces: vector spaces with complete translation-invariant metrics. LF-spaces are countable inductive limits of Fréchet spaces.
