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 (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

A Fréchet space X is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS,[1] meaning that every Cauchy sequence in X converges to some point in X (see footnote for more details).[2]

Important note: Not all authors require that a Fréchet space be locally convex (discussed below).

The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely, if the topology of a locally convex space X is induced by a translation-invariant complete metric then X is a Fréchet space.

Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local convexity requirement (such a space is today often called an "F-space").[1] The condition of locally convex was added later by Nicolas Bourbaki.[1] It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "F-space" and "Fréchet space" interchangeably. When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "F-space" and "Fréchet space" requires local convexity.[1]

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:

1. It is locally convex.[nb 1]
2. Its topology can be induced by a translation-invariant metric, i.e. a metric d: X × XR 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(v, u) < ε} is a subset of U.
3. Some (or equivalently, every) translation-invariant metric on X inducing the topology of X is complete.
• Assuming that the other two conditions are satisfied, this condition is equivalent to X being a complete topological vector space, meaning that X is a complete uniform space when it is endowed with its canonical uniformity (this canonical uniformly is independent of any metric on X and is defined entirely in terms of vector subtraction and X 's neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on X is identical to this canonical uniformity).

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:

• it is a Hausdorff 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 : ||v - u||k < ε for all kK} is a subset of U.
• it is complete with respect to the family of semi-norms

A family ${\displaystyle {\mathcal {P}}}$  of seminorms on ${\displaystyle X}$  yields a Hausdorff topology if and only if[3]

${\displaystyle \bigcap _{\|\,\cdot \,\|\in {\mathcal {P}}}\{x\in X:\|x\|=0\}=\{0\}.}$

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.

Comparison to Banach spaces

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 F-norm (the F stands for Fréchet).

Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential 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.

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,

${\displaystyle \|x\|\geq 0}$
${\displaystyle \|x+y\|\leq \|x\|+\|y\|}$
${\displaystyle \|c\cdot x\|=|c|\|x\|}$

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 xX and ǁxǁk = 0 for all k ≥ 0, then x = 0;
• if (xn) is a sequence in X which is Cauchy with respect to each semi-norm ǁ ⋅ ǁk, then there exists xX such that (xn) converges to x with respect to each semi-norm ǁ ⋅ ǁk.

Then the topology induced by these seminorms (as explained above) 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

${\displaystyle d(x,y)=\sum _{k=0}^{\infty }2^{-k}{\frac {\|x-y\|_{k}}{1+\|x-y\|_{k}}}\qquad x,y\in X.}$

The function uu/(1+u) maps [0, ∞) monotonically to [0, 1), and so the above definition ensures that d(x, y) is "small" if and only if there exists K "large" such that ǁx - yǁk is "small" for k = 0, …, K.

Theorem[4] (de Wilde 1978) — A topological vector space X is a Fréchet space if and only if it is a webbed space and a Baire space.

Examples

• 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 differentiable functions ƒ: [0,1] → R becomes a Fréchet space with the seminorms
${\displaystyle \|f\|_{k}=\sup\{|f^{(k)}(x)|:x\in [0,1]\}}$
for every non-negative integer k. Here, ƒ(k) denotes the k-th derivative of ƒ, and ƒ(0) = ƒ.
In this Fréchet space, a sequence (ƒn) of functions converges towards the element ƒ of C([0, 1]) if and only if for every non-negative integer k, the sequence (${\displaystyle f_{n}^{(k)}}$ ) converges uniformly towards ƒ(k).
• The vector space C(R) of all infinitely differentiable functions ƒ: RR becomes a Fréchet space with the seminorms
${\displaystyle \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$
for all integers k, n ≥ 0.
• The vector space Cm(R) of all m-times continuously differentiable functions ƒ: RR becomes a Fréchet space with the seminorms
${\displaystyle \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in [-n,n]\}}$
for all integers n ≥ 0 and k=0, ...,m.
• Let H be the space of entire (everywhere holomorphic) functions on the complex plane. Then the family of seminorms
${\displaystyle \|f\|_{n}=\sup\{|f(z)|:|z|\leq n\}}$
makes H into a Fréchet space.
• Let H be the space of entire (everywhere holomorphic) functions of exponential type τ. Then the family of seminorms
${\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|}$
makes H into a Fréchet space.
• If M is a compact C-manifold and B is a Banach space, then the set C(M, B) of all infinitely-often differentiable functions ƒ: MB can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If M is a (not necessarily compact) 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(M, B) and C(M, B) are also Fréchet space in a natural manner.
As a special case, every smooth finite-dimensional complete manifold M can be made into such a nested union of compact subsets: equip it with a Riemannian metric g which induces a metric d(x, y), choose x in M, and let
${\displaystyle K_{n}=\{y\in M|d(x,y)\leq n\}\ .}$
Let X be a compact C-manifold and V a vector bundle over X. Let C(X, V) denote the space of smooth sections of V over X. Choose Riemannian metrics and connections, which are guaranteed to exist, on the bundles TX and V. If s is a section, denote its jth covariant derivative by Djs. Then
${\displaystyle \|s\|_{n}=\sum _{j=0}^{n}\sup _{x\in M}|D^{j}s|}$
(where |⋅| is the norm induced by the Riemannian metric) is a family of seminorms making C(M, V) into a Fréchet space.

Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the space Lp([0, 1]) 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([a,b]), C(X, V) with X compact, and H all admit norms, while ω and C(ℝ) 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 (TVS) is continuous.[5]

There exists a Fréchet space X having a bounded subset B and also a dense vector subspace M such that B is not contained in the closure (in X) of any bounded subset of M.[6]

If X is a non-normable Fréchet space on which there exists a continuous norm, then X contains a closed vector subspace that has no topological complement.[7]

All metrizable Montel spaces are separable.[8] A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual converges is [[Strong dual space|strongly convergent}}.[8]

The following theorem implies that if X is a locally convex space then the topology of X can be a defined by a family of continuous norms on X (a norm is an injective seminorm) if and only if there exists at least one continuous norm on X.[9]

Theorem[9] — Let X be a Fréchet space over the field 𝕂. Then the following are equivalent:

1. X is not admit a continuous norm (that is, any continuous seminorm on X can not be a norm).
2. X contains a vector subspace that is TVS-isomorphic to 𝕂.
3. X contains a complemented vector subspace that is TVS-isomorphic to 𝕂.

Anderson–Kadec theorem — Every infinite-dimensional, separable real Fréchet space is homeomorphic to , the Cartesian product of countably many copies of the real line .

Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear.

Eidelheit theorem — A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to .

Differentiation of functions

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 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

${\displaystyle D(P)(x)(h)=\lim _{t\to 0}\,{\frac {1}{t}}{\Big (}P(x+th)-P(x){\Big )}}$

exists. We call P continuously differentiable in U if

${\displaystyle D(P):U\times X\to Y}$

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(ƒ) = ƒ′ is itself infinitely differentiable. The first derivative is given by

${\displaystyle D(P)(f)(h)=h'}$

for any two elements ƒ 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 : UY is a continuously differentiable function, then the differential equation

${\displaystyle x'(t)=P(x(t)),\quad x(0)=x_{0}\in U}$

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 (just like ordinary manifolds are defined as spaces that locally look like Euclidean space 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 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 (C) mappings γ : S1G, multiplied pointwise by 1 γ2)(t) = γ1(t) γ2(t).[10][11]

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

1. ^ Some authors do not make this requirement.

Notes

1. ^ a b c d Narici & Beckenstein 2011, p. 93.
2. ^ Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence x = (xm)
m=1
in a TVS X is Cauchy if and only if for all neighborhoods U of 0 in X, xm - xnU whenever m and n are sufficiently large. Note that this definition of a Cauchy sequence does not depend on any particular metric and doesn't even require that X be metrizable.
3. ^ Conway 1990, Chapter 4.
4. ^ Narici & Beckenstein 2011, p. 472.
5. ^ Treves 2006, p. 142.
6. ^ Wilansky 2013, p. 57.
7. ^ Schaefer & Wolff 1999, pp. 190-202.
8. ^ a b Schaefer 1999, pp. 194-195.
9. ^ a b Jarchow 1981, pp. 129-130.
10. ^ Sergeev 2010
11. ^ Pressley & Segal 1986

References

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