(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.)

Fréchet spaces can be defined in two equivalent ways. The first employs a translationinvariant metric, the second a countable family of seminorms.
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 translationinvariant 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 seminorms if and only if it converges to x with respect to each of the given seminorms.
The vector space C^{∞}([0,1]) of all infinitely often differentiable functions f : [0,1] → R becomes a Fréchet space with the seminorms
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 of all sequences of real numbers becomes a Fréchet space if we define the kth seminorm of a sequence to be the absolute value of the kth element of the sequence. Convergence in this Fréchet space is equivalent to elementwise 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 : 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
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
If P : U → Y is a continuously differentiable function, then the differential equation
The inverse function theorem[?] is not true in Fréchet spaces; a partial substitute is the NashMoser 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 R^{n}), 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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