Fréchet space


In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces.
A Fréchet space is a locally convex metrizable topological vector space that is complete as a TVS, meaning that every Cauchy sequence in converges to some point in .
The topology of every Fréchet space is induced by some translation-invariant complete metric.
Conversely, if the topology of a locally convex space is induced by a translation-invariant complete metric then is a Fréchet space.
In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm.
The topology of a Fréchet space does, however, arise from both a total paranorm and an -norm.
Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold.
Spaces of infinitely differentiable functions are typical examples of Fréchet spaces.
Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a completely metrizable topological vector space, without the local convexity requirement.
The condition of locally convex was added later by Nicolas Bourbaki.
It's important to note that some authors use "F-space" to mean a Fréchet space while others do not require that a "Fréchet space" be locally convex.
While reading mathematical literature, it is recommended that a reader always check whether the article's definition of "-space" and "Fréchet space" requires local convexity.

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:
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:
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:
Then the topology induced by these seminorms turns X into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete.
A translation-invariant complete metric inducing the same topology on X can then be defined by
The function uu/ monotonically to 0, 1), and so the above definition ensures that d is "small" if and only if there exists K "large" such that ǁx - yǁk is "small" for.

Examples

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: UY is a function, xU and hX.
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 and define the higher derivatives of P in this fashion.
The derivative operator P : CC 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 : UY 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: MM 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 : S1G, 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.

Remarks