{"history":[],"article":{"id":"9d3f11f5-11a1-4e81-92d1-ed5d1a19aba8","title":"Fréchet space","slug":"fr-chet-space","content":":::summary\n:::\n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), **Fréchet spaces**, named after [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet), are special [topological vector spaces](https://bliptext.com/articles/topological-vector-spaces).\nThey are generalizations of [Banach spaces](https://bliptext.com/articles/banach-spaces) ([normed vector spaces](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-vector-spaces) that are [complete](https://bliptext.com/articles/complete-space) with respect to the [metric](https://bliptext.com/articles/metric-mathematics) induced by the [norm](https://bliptext.com/articles/norm-mathematics)).\nAll [Banach](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) are Fréchet spaces.\nSpaces of [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) [functions](https://bliptext.com/articles/function-mathematics) are typical examples of Fréchet spaces, many of which are typically Banach spaces.\nA Fréchet space X is defined to be a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) that is [complete as a TVS](https://bliptext.com/articles/complete-topological-vector-space), meaning that every [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence) in X converges to some point in X (see footnote for more details).\nThe topology of every Fréchet space is induced by some [translation-invariant](https://bliptext.com/articles/translation-invariant) [complete metric](https://bliptext.com/articles/complete-metric-space).\nConversely, if the topology of a locally convex space X is induced by a translation-invariant complete metric then X is a Fréchet space.\n[Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) was the first to use the term \"[Banach space](https://bliptext.com/articles/banach-space)\" and [Banach](https://bliptext.com/articles/stefan-banach) in turn then coined the term \"[Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) space\" to mean a [complete](https://bliptext.com/articles/complete-topological-vector-space) [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space), without the local convexity requirement (such a space is today often called an \"[F-space](https://bliptext.com/articles/f-space)\").\nThe local convexity requirement was added later by [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki).\nIt'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.\nMoreover, some authors even use \"F-space\" and \"Fréchet space\" interchangeably.\nWhen reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of \"-space\" and \"Fréchet space\" requires local convexity.\n\n# Definitions\nFréchet spaces can be defined in two equivalent ways: the first employs a [translation-invariant](https://bliptext.com/articles/translation-invariant) [metric](https://bliptext.com/articles/metric-mathematics), the second a [countable](https://bliptext.com/articles/countable) family of [seminorms](https://bliptext.com/articles/seminorm).\n\n## Invariant metric definition\nA topological vector space X is a **Fréchet space** if and only if it satisfies the following three properties:\n

It is [locally convex](https://bliptext.com/articles/locally-convex).\n
Some (or equivalently, every) translation-invariant metric on X inducing the topology of X is [complete](https://bliptext.com/articles/complete-metric).\n

\nNote there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.\n\n## Countable family of seminorms definition\nThe 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:\nA family \\mathcal P of seminorms on X yields a Hausdorff topology if and only if\nA sequence 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.\n\n## As webbed Baire spaces\n\n## Comparison to Banach spaces\nIn contrast to [Banach spaces](https://bliptext.com/articles/banach-space), the complete translation-invariant metric need not arise from a norm.\nThe topology of a Fréchet space does, however, arise from both a [total paranorm](https://bliptext.com/articles/paranorm) and an [-norm](https://bliptext.com/articles/f-seminorm) (the stands for Fréchet).\nEven though the [topological structure](https://bliptext.com/articles/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](https://bliptext.com/articles/open-mapping-theorem-functional-analysis), the [closed graph theorem](https://bliptext.com/articles/closed-graph-theorem), and the [Banach–Steinhaus theorem](https://bliptext.com/articles/banach-steinhaus-theorem), still hold.\n\n# Constructing Fréchet spaces\nRecall that a seminorm \\|\\cdot\\| is a function from a vector space X to the real numbers satisfying three properties.\nFor all x, y \\in X and all scalars c,\nIf, then \\|\\cdot\\| is in fact a norm.\nHowever, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:\nTo construct a Fréchet space, one typically starts with a vector space X and defines a countable family of seminorms \\|\\cdot\\|_k on X with the following two properties:\nThen 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.\nA translation-invariant complete metric inducing the same topology on X can then be defined by\nThe function maps [0, \\infty) 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 \\|x - y\\|_k is \"small\" for\n\n# Examples\n\n## From pure functional analysis\n\n## From smooth manifolds\n

The [vector space](https://bliptext.com/articles/vector-space) of all infinitely differentiable functions becomes a Fréchet space with the seminorms\nfor every non-negative integer k. Here, f^{(k)} denotes the k-th derivative of f, and In this Fréchet space, a sequence of functions [converges](https://bliptext.com/articles/limit-mathematics) towards the element if and only if for every non-negative integer k \\geq 0, the sequence [converges](https://bliptext.com/articles/limit-mathematics) uniformly.
The vector space of all infinitely differentiable functions becomes a Fréchet space with the seminorms\nfor all integers Then, a sequence of functions converges if and only if for every the sequences [converge compactly](https://bliptext.com/articles/compact-convergence).
The vector space C^m(\\R) of all m-times continuously differentiable functions becomes a Fréchet space with the seminorms\nfor all integers n \\geq 0 and
If M is a [compact](https://bliptext.com/articles/compact-space) C^{\\infty}-[manifold](https://bliptext.com/articles/manifold) and B is a [Banach space](https://bliptext.com/articles/banach-space), then the set of all infinitely-often differentiable functions f : M \\to B can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives.\nIf M is a (not necessarily compact) C^{\\infty}-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 are also Fréchet space in a natural manner.\nAs a special case, every smooth finite-dimensional M can be made into such a nested union of compact subsets: equip it with a [Riemannian metric](https://bliptext.com/articles/riemannian-metric) g which induces a metric d(x, y), choose x \\in M, and let\nLet X be a compact C^{\\infty}-[manifold](https://bliptext.com/articles/manifold) andV a [vector bundle](https://bliptext.com/articles/vector-bundle) over X. Let denote the space of smooth sections of V over X. Choose Riemannian metrics g and connections D, which are guaranteed to exist, on the bundles TX and V. If s is a section, denote its jth covariant derivative by D^j s. Then\n(where is the norm induced by the Riemannian metric g) is a family of seminorms making into a Fréchet space.

\n\n## From holomorphicity\n

Let H be the space of entire (everywhere [holomorphic](https://bliptext.com/articles/holomorphic-function)) functions on the complex plane.\nThen the family of seminorms\nmakes H into a Fréchet space.
Let H be the space of entire (everywhere holomorphic) functions of [exponential type](https://bliptext.com/articles/exponential-type) \\tau.\nThen the family of seminorms\nmakes H into a Fréchet space.

\nNot all vector spaces with complete translation-invariant metrics are Fréchet spaces.\nAn example is the [space L^p([0, 1])](https://bliptext.com/articles/lp-space) with p < 1.\nAlthough this space fails to be locally convex, it is an [F-space](https://bliptext.com/articles/f-space).\n\n# Properties and further notions\nIf a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them.\nA Banach space, with X compact, and H all admit norms, while \\R^{\\omega} and C(\\R) do not.\nA closed subspace of a Fréchet space is a Fréchet space.\nA quotient of a Fréchet space by a closed subspace is a Fréchet space.\nThe direct sum of a finite number of Fréchet spaces is a Fréchet space.\nA product of [countably many](https://bliptext.com/articles/countable-set) Fréchet spaces is always once again a Fréchet space.\nHowever, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all for at most countably many of them are trivial (that is, have dimension 0).\nConsequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis).\nSo for example, if is any set and X is any non-trivial Fréchet space (such as X = \\R for instance), then the product is a Fréchet space if and only if I is a countable set.\nSeveral important tools of functional analysis which are based on the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem) remain true in Fréchet spaces; examples are the [closed graph theorem](https://bliptext.com/articles/closed-graph-theorem) and the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis).\nThe [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) implies that if are topologies on X that make both (X, \\tau) and into [complete](https://bliptext.com/articles/complete-topological-vector-space) [metrizable TVSs](https://bliptext.com/articles/metrizable-topological-vector-space) (such as Fréchet spaces) and if one topology is [finer or coarser](https://bliptext.com/articles/comparison-of-topologies) than the other then they must be equal (that is, if ).\nEvery [bounded](https://bliptext.com/articles/bounded-linear-map) linear operator from a Fréchet space into another [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is continuous.\nThere exists a Fréchet space X having a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subset of M.\nAll Fréchet spaces are [stereotype spaces](https://bliptext.com/articles/stereotype-space).\nIn the theory of stereotype spaces Fréchet spaces are dual objects to [Brauner spaces](https://bliptext.com/articles/brauner-space).\nAll [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [Montel spaces](https://bliptext.com/articles/montel-space) are [separable](https://bliptext.com/articles/separable-space).\nA [separable](https://bliptext.com/articles/separable-space) Fréchet space is a Montel space if and only if each [weak-* convergent](https://bliptext.com/articles/weak-topology) sequence in its continuous dual converges is [strongly convergent](https://bliptext.com/articles/strong-dual-space).\nThe [strong dual space](https://bliptext.com/articles/strong-dual-space) of a Fréchet space (and more generally, of any metrizable locally convex space) X is a [DF-space](https://bliptext.com/articles/df-space).\nThe strong dual of a DF-space is a Fréchet space.\nThe strong dual of a [reflexive](https://bliptext.com/articles/reflexive-space) Fréchet space is a [bornological space](https://bliptext.com/articles/bornological-space) and a [Ptak space](https://bliptext.com/articles/ptak-space).\nEvery Fréchet space is a Ptak space.\nThe strong bidual (that is, the [strong dual space](https://bliptext.com/articles/strong-dual-space) of the [strong dual space](https://bliptext.com/articles/strong-dual-space)) of a metrizable locally convex space is a Fréchet space.\n\n## Norms and normability\nIf X is a locally convex space then the topology of X can be a defined by a family of continuous on X (a **[norm](https://bliptext.com/articles/norm-mathematics)** is a [positive-definite](https://bliptext.com/articles/positive-definite-functional) semi**[norm](https://bliptext.com/articles/norm-mathematics)**) if and only if there exists continuous on X. Even if a Fréchet space has a topology that is defined by a (countable) family of (all **[norm](https://bliptext.com/articles/norm-mathematics)**s are also semi**[norm](https://bliptext.com/articles/norm-mathematics)**s), then it may nevertheless still fail to be **[norm](https://bliptext.com/articles/norm-mathematics)**able space (meaning that its topology can not be defined by any single **[norm](https://bliptext.com/articles/norm-mathematics)**).\nThe [space of all sequences](https://bliptext.com/articles/space-of-real-sequences) (with the product topology) is a Fréchet space.\nThere does not exist any Hausdorff [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) topology on that is [strictly coarser](https://bliptext.com/articles/comparison-of-topologies) than this product topology.\nThe space is not [normable](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)able-space), which means that its topology can not be defined by any [norm](https://bliptext.com/articles/norm-mathematics).\nAlso, there does not exist [continuous](https://bliptext.com/articles/continuous-function) norm on In fact, as the following theorem shows, whenever X is a Fréchet space on which there does not exist any [continuous](https://bliptext.com/articles/continuous-function) norm, then this is due entirely to the presence of as a subspace.\nIf 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](https://bliptext.com/articles/topological-complement).\nA metrizable [locally convex](https://bliptext.com/articles/locally-convex) space is [normable](https://bliptext.com/articles/normable-space) if and only if its [strong dual space](https://bliptext.com/articles/strong-dual-space) is a [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) [locally convex](https://bliptext.com/articles/locally-convex) space.\nIn particular, if a locally convex metrizable space X (such as a Fréchet space) is normable (which can only happen if X is infinite dimensional) then its [strong dual space](https://bliptext.com/articles/strong-dual-space) is not a [Fréchet–Urysohn space](https://bliptext.com/articles/fr-chet-urysohn-space) and consequently, this [complete](https://bliptext.com/articles/complete-topological-vector-space) Hausdorff locally convex space is also neither metrizable nor normable.\nThe [strong dual space](https://bliptext.com/articles/strong-dual-space) of a Fréchet space (and more generally, of [bornological spaces](https://bliptext.com/articles/bornological-space) such as metrizable TVSs) is always a [complete TVS](https://bliptext.com/articles/complete-topological-vector-space) and so like any [complete TVS](https://bliptext.com/articles/complete-topological-vector-space), it is [normable](https://bliptext.com/articles/normable-space) if and only if its topology can be induced by a [complete norm](https://bliptext.com/articles/complete-norm) (that is, if and only if it can be made into a [Banach space](https://bliptext.com/articles/banach-space) that has the same topology).\nIf X is a Fréchet space then X is [normable](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)able-space) if (and only if) there exists a complete [norm](https://bliptext.com/articles/norm-mathematics) on its continuous dual space X' such that the [norm](https://bliptext.com/articles/norm-mathematics) induced topology on X' is [finer](https://bliptext.com/articles/comparison-of-topologies) than the weak-* topology.\nConsequently, if a Fréchet space is normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.\n\n## Anderson–Kadec theorem\nNote that the homeomorphism described in the Anderson–Kadec theorem is necessarily linear.\n\n# Differentiation of functions\nIf X and Y are Fréchet spaces, then the space L(X,Y) consisting of all [continuous](https://bliptext.com/articles/continuous-function-topology) [linear maps](https://bliptext.com/articles/linear-operator) from X to Y is a Fréchet space in any natural manner.\nThis 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](https://bliptext.com/articles/gateaux-derivative):\nSuppose U is an open subset of a Fréchet space X, P : U \\to Y is a function valued in a Fréchet space Y, x \\in U and h \\in X. The map P is **differentiable at x in the direction h** if the [limit](https://bliptext.com/articles/limit-mathematics)\nexists.\nThe map P is said to be **continuously differentiable** in U if the map\nis continuous.\nSince the [product](https://bliptext.com/articles/product-topology) 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.\nThe derivative operator defined by P(f) = f' is itself infinitely differentiable.\nThe first derivative is given by\nfor any two elements\nThis is a major advantage of the Fréchet space over the Banach space C^k([0, 1]) for finite k.\nIf P : U \\to Y is a continuously differentiable function, then the [differential equation](https://bliptext.com/articles/differential-equation)\nneed not have any solutions, and even if does, the solutions need not be unique.\nThis is in stark contrast to the situation in Banach spaces.\nIn general, the [inverse function theorem](https://bliptext.com/articles/inverse-function-theorem) is not true in Fréchet spaces, although a partial substitute is the [Nash–Moser theorem](https://bliptext.com/articles/nash-moser-theorem).\n\n# Fréchet manifolds and Lie groups\nOne 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](https://bliptext.com/articles/euclidean-space) \\R^n), and one can then extend the concept of [Lie group](https://bliptext.com/articles/lie-group) to these manifolds.\nThis is useful because for a given (ordinary) compact C^{\\infty} manifold M, the set of all C^{\\infty} [diffeomorphisms](https://bliptext.com/articles/diffeomorphism) f : M \\to 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](https://bliptext.com/articles/lie-algebra) and Lie groups remain valid in this setting.\nAnother important example of a Fréchet Lie group is the loop group of a compact Lie group G, the smooth (C^{\\infty}) mappings multiplied pointwise by\n\n# Generalizations\nIf we drop the requirement for the space to be locally convex, we obtain [F-spaces](https://bliptext.com/articles/f-space): vector spaces with complete translation-invariant metrics.\n[LF-spaces](https://bliptext.com/articles/lf-space) are countable inductive limits of Fréchet spaces.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:48:56.308Z","updated_at":"2025-01-25T18:48:56.308Z","search_vector":"'fréchet':1 'space':2","search_content":" \n \n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), **Fréchet spaces**, named after [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet), are special [topological vector spaces](https://bliptext.com/articles/topological-vector-spaces).\nThey are generalizations of [Banach spaces](https://bliptext.com/articles/banach-spaces) ([[norm](https://bliptext.com/articles/norm-mathematics)ed vector spaces](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)ed-vector-spaces) that are [complete](https://bliptext.com/articles/complete-space) with respect to the [metric](https://bliptext.com/articles/metric-mathematics) induced by the [norm](https://bliptext.com/articles/norm-mathematics)).\nAll [Banach](https://bliptext.com/articles/banach-space) and [Hilbert spaces](https://bliptext.com/articles/hilbert-space) are Fréchet spaces.\nSpaces of [infinitely differentiable](https://bliptext.com/articles/infinitely-differentiable) [functions](https://bliptext.com/articles/function-mathematics) are typical examples of Fréchet spaces, many of which are typically Banach spaces.\nA Fréchet space X is defined to be a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) that is [complete as a TVS](https://bliptext.com/articles/complete-topological-vector-space), meaning that every [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence) in X converges to some point in X (see footnote for more details).\nThe topology of every Fréchet space is induced by some [translation-invariant](https://bliptext.com/articles/translation-invariant) [complete metric](https://bliptext.com/articles/complete-metric-space).\nConversely, if the topology of a locally convex space X is induced by a translation-invariant complete metric then X is a Fréchet space.\n[Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) was the first to use the term \"[[Banach](https://bliptext.com/articles/stefan-banach) space](https://bliptext.com/articles/banach-space)\" and [Banach](https://bliptext.com/articles/stefan-banach) in turn then coined the term \"[Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) space\" to mean a [complete](https://bliptext.com/articles/complete-topological-vector-space) [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space), without the local convexity requirement (such a space is today often called an \"[F-space](https://bliptext.com/articles/f-space)\").\nThe local convexity requirement was added later by [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki).\nIt'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.\nMoreover, some authors even use \"F-space\" and \"Fréchet space\" interchangeably.\nWhen reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of \"-space\" and \"Fréchet space\" requires local convexity.\n\n# Definitions\nFréchet spaces can be defined in two equivalent ways: the first employs a [translation-invariant](https://bliptext.com/articles/translation-invariant) [metric](https://bliptext.com/articles/metric-mathematics), the second a [countable](https://bliptext.com/articles/countable) family of [seminorms](https://bliptext.com/articles/seminorm).\n\n## Invariant metric definition\nA topological vector space X is a **Fréchet space** if and only if it satisfies the following three properties:\n \n It is [locally convex](https://bliptext.com/articles/locally-convex).\n \n**** ****I****t****s**** ****t****o****p****o****l****o****g****y**** **** ****b****e**** ****i****n****d****u****c****e****d**** ****b****y**** ****a**** ****t****r****a****n****s****l****a****t****i****o****n****-****i****n****v****a****r****i****a****n****t**** ****m****e****t****r****i****c****,**** ****t****h****a****t**** ****i****s****,**** ****a**** ****m****e****t****r****i****c**** ****s****u****c****h**** ****t****h****a****t**** **** ****f****o****r**** ****a****l****l**** **** ****T****h****i****s**** ****m****e****a****n****s**** ****t****h****a****t**** ****a**** ****s****u****b****s****e****t**** ****U**** ****o****f**** ****X**** ****i****s**** ****o****p****e****n**** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****f****o****r**** ****e****v****e****r****y**** ****u**** ****\\****i****n**** ****U**** ****t****h****e****r****e**** ****e****x****i****s****t****s**** ****a****n**** ****r**** ****>**** ****0**** ****s****u****c****h**** ****t****h****a****t**** **** ****i****s**** ****a**** ****s****u****b****s****e****t**** ****o****f**** ****U****.**** ****\n Some (or equivalently, every) translation-invariant metric on X inducing the topology of X is [complete](https://bliptext.com/articles/complete-metric).\n \nNote there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.\n\n## Countable family of seminorms definition\nThe 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:\nA family \\mathcal P of seminorms on X yields a Hausdorff topology if and only if\nA sequence 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.\n\n## As webbed Baire spaces\n\n## Comparison to Banach spaces\nIn contrast to [Banach spaces](https://bliptext.com/articles/banach-space), the complete translation-invariant metric need not arise from a norm.\nThe topology of a Fréchet space does, however, arise from both a [total paranorm](https://bliptext.com/articles/paranorm) and an [-norm](https://bliptext.com/articles/f-seminorm) (the stands for Fréchet).\nEven though the [topological structure](https://bliptext.com/articles/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](https://bliptext.com/articles/open-mapping-theorem-functional-analysis), the [closed graph theorem](https://bliptext.com/articles/closed-graph-theorem), and the [Banach–Steinhaus theorem](https://bliptext.com/articles/banach-steinhaus-theorem), still hold.\n\n# Constructing Fréchet spaces\nRecall that a seminorm \\|\\cdot\\| is a function from a vector space X to the real numbers satisfying three properties.\nFor all x, y \\in X and all scalars c,\nIf, then \\|\\cdot\\| is in fact a norm.\nHowever, seminorms are useful in that they enable us to construct Fréchet spaces, as follows:\nTo construct a Fréchet space, one typically starts with a vector space X and defines a countable family of seminorms \\|\\cdot\\|_k on X with the following two properties:\nThen 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.\nA translation-invariant complete metric inducing the same topology on X can then be defined by\nThe function maps [0, \\infty) 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 \\|x - y\\|_k is \"small\" for\n\n# Examples\n\n## From pure functional analysis\n\n## From smooth manifolds\n \n The [vector space](https://bliptext.com/articles/vector-space) of all infinitely differentiable functions becomes a Fréchet space with the seminorms\nfor every non-negative integer k. Here, f^{(k)} denotes the k-th derivative of f, and In this Fréchet space, a sequence of functions [converges](https://bliptext.com/articles/limit-mathematics) towards the element if and only if for every non-negative integer k \\geq 0, the sequence [converges](https://bliptext.com/articles/limit-mathematics) uniformly. \n The vector space of all infinitely differentiable functions becomes a Fréchet space with the seminorms\nfor all integers Then, a sequence of functions converges if and only if for every the sequences [converge compactly](https://bliptext.com/articles/compact-convergence). \n The vector space C^m(\\R) of all m-times continuously differentiable functions becomes a Fréchet space with the seminorms\nfor all integers n \\geq 0 and \n If M is a [compact](https://bliptext.com/articles/compact-space) C^{\\infty}-[manifold](https://bliptext.com/articles/manifold) and B is a [Banach space](https://bliptext.com/articles/banach-space), then the set of all infinitely-often differentiable functions f : M \\to B can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives.\nIf M is a (not necessarily compact) C^{\\infty}-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 are also Fréchet space in a natural manner.\nAs a special case, every smooth finite-dimensional M can be made into such a nested union of compact subsets: equip it with a [Riemannian metric](https://bliptext.com/articles/riemannian-metric) g which induces a metric d(x, y), choose x \\in M, and let\nLet X be a compact C^{\\infty}-[manifold](https://bliptext.com/articles/manifold) andV a [vector bundle](https://bliptext.com/articles/vector-bundle) over X. Let denote the space of smooth sections of V over X. Choose Riemannian metrics g and connections D, which are guaranteed to exist, on the bundles TX and V. If s is a section, denote its jth covariant derivative by D^j s. Then\n(where is the norm induced by the Riemannian metric g) is a family of seminorms making into a Fréchet space. \n \n\n## From holomorphicity\n \n Let H be the space of entire (everywhere [holomorphic](https://bliptext.com/articles/holomorphic-function)) functions on the complex plane.\nThen the family of seminorms\nmakes H into a Fréchet space. \n Let H be the space of entire (everywhere holomorphic) functions of [exponential type](https://bliptext.com/articles/exponential-type) \\tau.\nThen the family of seminorms\nmakes H into a Fréchet space. \n \nNot all vector spaces with complete translation-invariant metrics are Fréchet spaces.\nAn example is the [space L^p([0, 1])](https://bliptext.com/articles/lp-space) with p < 1.\nAlthough this space fails to be locally convex, it is an [F-space](https://bliptext.com/articles/f-space).\n\n# Properties and further notions\nIf a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them.\nA Banach space, with X compact, and H all admit norms, while \\R^{\\omega} and C(\\R) do not.\nA closed subspace of a Fréchet space is a Fréchet space.\nA quotient of a Fréchet space by a closed subspace is a Fréchet space.\nThe direct sum of a finite number of Fréchet spaces is a Fréchet space.\nA product of [countably many](https://bliptext.com/articles/countable-set) Fréchet spaces is always once again a Fréchet space.\nHowever, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all for at most countably many of them are trivial (that is, have dimension 0).\nConsequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis).\nSo for example, if is any set and X is any non-trivial Fréchet space (such as X = \\R for instance), then the product is a Fréchet space if and only if I is a countable set.\nSeveral important tools of functional analysis which are based on the [Baire category theorem](https://bliptext.com/articles/baire-category-theorem) remain true in Fréchet spaces; examples are the [closed graph theorem](https://bliptext.com/articles/closed-graph-theorem) and the [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis).\nThe [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) implies that if are topologies on X that make both (X, \\tau) and into [complete](https://bliptext.com/articles/complete-topological-vector-space) [metrizable TVSs](https://bliptext.com/articles/metrizable-topological-vector-space) (such as Fréchet spaces) and if one topology is [finer or coarser](https://bliptext.com/articles/comparison-of-topologies) than the other then they must be equal (that is, if ).\nEvery [bounded](https://bliptext.com/articles/bounded-linear-map) linear operator from a Fréchet space into another [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is continuous.\nThere exists a Fréchet space X having a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subset B and also a dense vector subspace M such that B is contained in the closure (in X) of any [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) subset of M.\nAll Fréchet spaces are [stereotype spaces](https://bliptext.com/articles/stereotype-space).\nIn the theory of stereotype spaces Fréchet spaces are dual objects to [Brauner spaces](https://bliptext.com/articles/brauner-space).\nAll [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [Montel spaces](https://bliptext.com/articles/montel-space) are [separable](https://bliptext.com/articles/separable-space).\nA [separable](https://bliptext.com/articles/separable-space) Fréchet space is a Montel space if and only if each [weak-* convergent](https://bliptext.com/articles/weak-topology) sequence in its continuous dual converges is [strongly convergent](https://bliptext.com/articles/strong-dual-space).\nThe [strong dual space](https://bliptext.com/articles/strong-dual-space) of a Fréchet space (and more generally, of any metrizable locally convex space) X is a [DF-space](https://bliptext.com/articles/df-space).\nThe strong dual of a DF-space is a Fréchet space.\nThe strong dual of a [reflexive](https://bliptext.com/articles/reflexive-space) Fréchet space is a [bornological space](https://bliptext.com/articles/bornological-space) and a [Ptak space](https://bliptext.com/articles/ptak-space).\nEvery Fréchet space is a Ptak space.\nThe strong bidual (that is, the [strong dual space](https://bliptext.com/articles/strong-dual-space) of the [strong dual space](https://bliptext.com/articles/strong-dual-space)) of a metrizable locally convex space is a Fréchet space.\n\n## Norms and normability\nIf X is a locally convex space then the topology of X can be a defined by a family of continuous on X (a **[norm](https://bliptext.com/articles/norm-mathematics)** is a [positive-definite](https://bliptext.com/articles/positive-definite-functional) semi**[norm](https://bliptext.com/articles/norm-mathematics)**) if and only if there exists continuous on X. Even if a Fréchet space has a topology that is defined by a (countable) family of (all **[norm](https://bliptext.com/articles/norm-mathematics)**s are also semi**[norm](https://bliptext.com/articles/norm-mathematics)**s), then it may nevertheless still fail to be **[norm](https://bliptext.com/articles/norm-mathematics)**able space (meaning that its topology can not be defined by any single **[norm](https://bliptext.com/articles/norm-mathematics)**).\nThe [space of all sequences](https://bliptext.com/articles/space-of-real-sequences) (with the product topology) is a Fréchet space.\nThere does not exist any Hausdorff [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) topology on that is [strictly coarser](https://bliptext.com/articles/comparison-of-topologies) than this product topology.\nThe space is not [[norm](https://bliptext.com/articles/norm-mathematics)able](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)able-space), which means that its topology can not be defined by any [norm](https://bliptext.com/articles/norm-mathematics).\nAlso, there does not exist [continuous](https://bliptext.com/articles/continuous-function) norm on In fact, as the following theorem shows, whenever X is a Fréchet space on which there does not exist any [continuous](https://bliptext.com/articles/continuous-function) norm, then this is due entirely to the presence of as a subspace.\nIf 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](https://bliptext.com/articles/topological-complement).\nA metrizable [locally convex](https://bliptext.com/articles/locally-convex) space is [normable](https://bliptext.com/articles/normable-space) if and only if its [strong dual space](https://bliptext.com/articles/strong-dual-space) is a [Fréchet–Urysohn](https://bliptext.com/articles/fr-chet-urysohn-space) [locally convex](https://bliptext.com/articles/locally-convex) space.\nIn particular, if a locally convex metrizable space X (such as a Fréchet space) is normable (which can only happen if X is infinite dimensional) then its [strong dual space](https://bliptext.com/articles/strong-dual-space) is not a [Fréchet–Urysohn space](https://bliptext.com/articles/fr-chet-urysohn-space) and consequently, this [complete](https://bliptext.com/articles/complete-topological-vector-space) Hausdorff locally convex space is also neither metrizable nor normable.\nThe [strong dual space](https://bliptext.com/articles/strong-dual-space) of a Fréchet space (and more generally, of [bornological spaces](https://bliptext.com/articles/bornological-space) such as metrizable TVSs) is always a [complete TVS](https://bliptext.com/articles/complete-topological-vector-space) and so like any [complete TVS](https://bliptext.com/articles/complete-topological-vector-space), it is [normable](https://bliptext.com/articles/normable-space) if and only if its topology can be induced by a [complete norm](https://bliptext.com/articles/complete-norm) (that is, if and only if it can be made into a [Banach space](https://bliptext.com/articles/banach-space) that has the same topology).\nIf X is a Fréchet space then X is [[norm](https://bliptext.com/articles/norm-mathematics)able](https://bliptext.com/articles/[norm](https://bliptext.com/articles/norm-mathematics)able-space) if (and only if) there exists a complete [norm](https://bliptext.com/articles/norm-mathematics) on its continuous dual space X' such that the [norm](https://bliptext.com/articles/norm-mathematics) induced topology on X' is [finer](https://bliptext.com/articles/comparison-of-topologies) than the weak-* topology.\nConsequently, if a Fréchet space is normable (which can only happen if it is infinite dimensional) then neither is its strong dual space.\n\n## Anderson–Kadec theorem\nNote that the homeomorphism described in the Anderson–Kadec theorem is necessarily linear.\n\n# Differentiation of functions\nIf X and Y are Fréchet spaces, then the space L(X,Y) consisting of all [continuous](https://bliptext.com/articles/continuous-function-topology) [linear maps](https://bliptext.com/articles/linear-operator) from X to Y is a Fréchet space in any natural manner.\nThis 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](https://bliptext.com/articles/gateaux-derivative):\nSuppose U is an open subset of a Fréchet space X, P : U \\to Y is a function valued in a Fréchet space Y, x \\in U and h \\in X. The map P is **differentiable at x in the direction h** if the [limit](https://bliptext.com/articles/limit-mathematics)\nexists.\nThe map P is said to be **continuously differentiable** in U if the map\nis continuous.\nSince the [product](https://bliptext.com/articles/product-topology) 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.\nThe derivative operator defined by P(f) = f' is itself infinitely differentiable.\nThe first derivative is given by\nfor any two elements\nThis is a major advantage of the Fréchet space over the Banach space C^k([0, 1]) for finite k.\nIf P : U \\to Y is a continuously differentiable function, then the [differential equation](https://bliptext.com/articles/differential-equation)\nneed not have any solutions, and even if does, the solutions need not be unique.\nThis is in stark contrast to the situation in Banach spaces.\nIn general, the [inverse function theorem](https://bliptext.com/articles/inverse-function-theorem) is not true in Fréchet spaces, although a partial substitute is the [Nash–Moser theorem](https://bliptext.com/articles/nash-moser-theorem).\n\n# Fréchet manifolds and Lie groups\nOne 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](https://bliptext.com/articles/euclidean-space) \\R^n), and one can then extend the concept of [Lie group](https://bliptext.com/articles/lie-group) to these manifolds.\nThis is useful because for a given (ordinary) compact C^{\\infty} manifold M, the set of all C^{\\infty} [diffeomorphisms](https://bliptext.com/articles/diffeomorphism) f : M \\to 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](https://bliptext.com/articles/lie-algebra) and Lie groups remain valid in this setting.\nAnother important example of a Fréchet Lie group is the loop group of a compact Lie group G, the smooth (C^{\\infty}) mappings multiplied pointwise by\n\n# Generalizations\nIf we drop the requirement for the space to be locally convex, we obtain [F-spaces](https://bliptext.com/articles/f-space): vector spaces with complete translation-invariant metrics.\n[LF-spaces](https://bliptext.com/articles/lf-space) are countable inductive limits of Fréchet spaces.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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