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).

A Fréchet space $X$ is 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] 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. 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 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 (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 some 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. While reading mathematical literature, it is recommended that a reader always check whether the 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:

It is locally convex.[nb 1]
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(v, u) < ε} is a subset of U.
Any (hence every) translation-invariant metric inducing the topology is complete; in other words, X is a complete topological vector space.

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 k ≤ K} is a subset of U.
it is complete with respect to the family of semi-norms

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

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

A sequence (x_n) 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,

\|x\|\geq 0

\|x+y\|\leq \|x\|+\|y\|

\|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 ⋅ ǁ_k on X with the following two properties:

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

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

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

The function u → u/(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 - yǁ_k is "small" for $k = 0, \dots, K$ .

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

\|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 (ƒ_n) of functions converges towards the element ƒ of C^∞([0, 1]) if and only if for every non-negative integer k, the sequence (

f_{n}^{(k)}

) converges uniformly towards ƒ^(k).

The vector space C^∞(R) of all infinitely differentiable functions ƒ: R → R becomes a Fréchet space with the seminorms

\|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in [-n,n]\}

for all integers k, n ≥ 0.

The vector space C^m(R) of all m-times continuously differentiable functions ƒ: R → R becomes a Fréchet space with the seminorms

\|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

\|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

\|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: 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 (not necessarily compact) C^∞-manifold which admits a countable sequence K_n of compact subsets, so that every compact subset of M is contained in at least one K_n, then the spaces C^m(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

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 D^js. Then

\|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.

The space R^ω 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 space L^p([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 R^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]

Theorem[7] — Let $X$ be a Fréchet space over the field $𝕂$ . Then the following are equivalent:

$X$ is not normable.
$X$ contains a vector subspace that is TVS-isomorphic to $𝕂 ℕ$ .
$X$ contains a complemented vector subspace that is TVS-isomorphic to $𝕂 ℕ$ .

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: 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

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

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

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 C^k([0,1]) for finite k.

If P : U → Y is a continuously differentiable function, then the differential equation

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 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 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 (C^∞) mappings γ : S¹ → G, multiplied pointwise by (γ₁ γ₂)(t) = γ₁(t) γ₂(t).[8][9]

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.

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Remarks

Some authors do not make this requirement.

Notes

Narici 2011, p. 93.
Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence $x • = (x m) \infty m =1$ in a TVS $X$ is Cauchy if and only if for all neighborhoods $U$ of 0 in $X$ , $x m - x n \in U$ 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.
Conway 1990, Chapter 4.
Narici 2011, p. 472.
Treves 2006, p. 142.
Wilansky 2013, p. 57.
Jarchow 1981, pp. 129-130.
Sergeev 2010
Pressley & Segal 1986

References

"Fréchet space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Pressley, Andrew; Segal, Graeme (1986). Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. New York: Oxford University Press. ISBN 0-19-853535-X. MR 0900587.CS1 maint: ref=harv (link)
Robertson, Alex P.; Robertson, Wendy J. (January 1, 1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
Sergeev, Armen (2010). Kähler Geometry of Loop Spaces. Mathematical Society of Japan Memoirs. 23. World Scientific Publishing. ISBN 978-4-931469-60-0.CS1 maint: ref=harv (link)
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.CS1 maint: ref=harv (link)
Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.CS1 maint: ref=harv (link)
Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)

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[3] Some authors do not make this requirement.

[FOOTNOTENarici201193-1] Narici 2011, p. 93.

[2] Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence $x • = (x m) \infty m =1$ in a TVS $X$ is Cauchy if and only if for all neighborhoods $U$ of 0 in $X$ , $x m - x n \in U$ 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.

[4] Conway 1990, Chapter 4.

[FOOTNOTENarici2011472-5] Narici 2011, p. 472.

[FOOTNOTETreves2006142-6] Treves 2006, p. 142.

[FOOTNOTEWilansky201357-7] Wilansky 2013, p. 57.

[FOOTNOTEJarchow1981129-130-8] Jarchow 1981, pp. 129-130.

[9] Sergeev 2010

[10] Pressley & Segal 1986

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-Valued Hahn-Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property