Convenient vector space

In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition.

Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces,[Note 1] for any compatible topology on the spaces of continuous linear mappings.

Mappings between convenient vector spaces are smooth or  C ∞ {\displaystyle C^{\infty }} if they map smooth curves to smooth curves. This leads to a Cartesian closed category of smooth mappings between  c ∞ {\displaystyle c^{\infty }} -open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called convenient calculus. It is weaker than any other reasonable notion of differentiability, it is easy to apply, but there are smooth mappings which are not continuous (see Note 1). This type of calculus alone is not useful in solving equations[Note 2].

## The c∞-topology

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See also: Differentiable vector-valued functions from Euclidean space and Differentiation in Fréchet spaces

Let  E {\displaystyle E} be a locally convex vector space. A curve  c : R -> E {\displaystyle c:\mathbb {R} \to E} is called smooth or  C ∞ {\displaystyle C^{\infty }} if all derivatives exist and are continuous. Let  C ∞ ( R , E ) {\displaystyle C^{\infty }(\mathbb {R} ,E)} be the space of smooth curves. It can be shown that the set of smooth curves does not depend entirely on the locally convex topology of  E , {\displaystyle E,} only on its associated bornology (system of bounded sets); see [KM], 2.11. The final topologies with respect to the following sets of mappings into  E {\displaystyle E} coincide; see [KM], 2.13.

* C ∞ ( R , E ) . {\displaystyle C^{\infty }(\mathbb {R} ,E).}
* The set of all Lipschitz curves (so that  { c ( t ) − c ( s ) t − s : t ≠ s , | t | , | s | ≤ C } {\displaystyle \left\\{{\dfrac {c(t)-c(s)}{t-s}}:t\neq s{,}|t|,|s|\leq C\right\\}} is bounded in  E , {\displaystyle E,} for each  C {\displaystyle C} ).
* The set of injections  E B -> E {\displaystyle E_{B}\to E} where  B {\displaystyle B} runs through all bounded absolutely convex subsets in  E , {\displaystyle E,} and where  E B {\displaystyle E_{B}} is the linear span of  B {\displaystyle B} equipped with the Minkowski functional ‖ x ‖ B := inf { λ > 0 : x ∈ λ B } . {\displaystyle \|x\|_{B}:=\inf\\{\lambda >0:x\in \lambda B\\}.}
* The set of all Mackey convergent sequences x n -> x {\displaystyle x_{n}\to x} (there exists a sequence  0 < λ n -> ∞ {\displaystyle 0<\lambda _{n}\to \infty } with  λ n ( x n − x ) {\displaystyle \lambda _{n}\left(x_{n}-x\right)} bounded).

This topology is called the  c ∞ {\displaystyle c^{\infty }} -topology on  E {\displaystyle E} and we write  c ∞ E {\displaystyle c^{\infty }E} for the resulting topological space. In general (on the space  D {\displaystyle D} of smooth functions with compact support on the real line, for example) it is finer than the given locally convex topology, it is not a vector space topology, since addition is no longer jointly continuous. Namely, even  c ∞ ( D × D ) ≠ ( c ∞ D ) × ( c ∞ D ) . {\displaystyle c^{\infty }(D\times D)\neq \left(c^{\infty }D\right)\times \left(c^{\infty }D\right).} The finest among all locally convex topologies on  E {\displaystyle E} which are coarser than  c ∞ E {\displaystyle c^{\infty }E} is the bornologification of the given locally convex topology. If  E {\displaystyle E} is a Fréchet space, then  c ∞ E = E . {\displaystyle c^{\infty }E=E.}

## Convenient vector spaces

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A locally convex vector space  E {\displaystyle E} is said to be a convenient vector space if one of the following equivalent conditions holds (called  c ∞ {\displaystyle c^{\infty }} -completeness); see [KM], 2.14.

* For any  c ∈ C ∞ ( R , E ) {\displaystyle c\in C^{\infty }(\mathbb {R} ,E)} the (Riemann-) integral  ∫ 0 1 c ( t ) d t {\displaystyle \int _{0}^{1}c(t)\,dt} exists in  E {\displaystyle E} .
* Any Lipschitz curve in  E {\displaystyle E} is locally Riemann integrable.
* Any scalar wise C ∞ {\displaystyle C^{\infty }} curve is  C ∞ {\displaystyle C^{\infty }} : A curve  c : R -> E {\displaystyle c:\mathbb {R} \to E} is smooth if and only if the composition  λ ∘ c : t ↦ λ ( c ( t ) ) {\displaystyle \lambda \circ c:t\mapsto \lambda (c(t))} is in  C ∞ ( R , R ) {\displaystyle C^{\infty }(\mathbb {R} ,\mathbb {R} )} for all  λ ∈ E ∗ {\displaystyle \lambda \in E^{*}} where  E ∗ {\displaystyle E^{*}} is the dual of all continuous linear functionals on  E {\displaystyle E} .
* Equivalently, for all  λ ∈ E ′ {\displaystyle \lambda \in E'} , the dual of all bounded linear functionals.
* Equivalently, for all  λ ∈ V {\displaystyle \lambda \in V} , where  V {\displaystyle V} is a subset of  E ′ {\displaystyle E'} which recognizes bounded subsets in  E {\displaystyle E} ; see [KM], 5.22.
* Any Mackey-Cauchy-sequence (i.e.,  t n m ( x − x m ) -> 0 {\displaystyle t_{nm}(x-x_{m})\to 0} for some  t n m -> ∞ {\displaystyle t_{nm}\to \infty } in  R {\displaystyle \mathbb {R} } converges in  E {\displaystyle E} . This is visibly a mild completeness requirement.
* If  B {\displaystyle B} is bounded closed absolutely convex, then  E B {\displaystyle E_{B}} is a Banach space.
* If  f : R -> E {\displaystyle f:\mathbb {R} \to E} is scalar wise  Lip k {\displaystyle {\text{Lip}}^{k}} , then  f {\displaystyle f} is  Lip k {\displaystyle {\text{Lip}}^{k}} , for  k > 1 {\displaystyle k>1} .
* If  f : R -> E {\displaystyle f:\mathbb {R} \to E} is scalar wise  C ∞ {\displaystyle C^{\infty }} then  f {\displaystyle f} is differentiable at  0 {\displaystyle 0} .

Here a mapping  f : R -> E {\displaystyle f:\mathbb {R} \to E} is called  Lip k {\displaystyle {\text{Lip}}^{k}} if all derivatives up to order  k {\displaystyle k} exist and are Lipschitz, locally on  R {\displaystyle \mathbb {R} } .

## Smooth mappings

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Let  E {\displaystyle E} and  F {\displaystyle F} be convenient vector spaces, and let  U ⊆ E {\displaystyle U\subseteq E} be  c ∞ {\displaystyle c^{\infty }} -open. A mapping  f : U -> F {\displaystyle f:U\to F} is called smooth or  C ∞ {\displaystyle C^{\infty }} , if the composition  f ∘ c ∈ C ∞ ( R , F ) {\displaystyle f\circ c\in C^{\infty }(\mathbb {R} ,F)} for all  c ∈ C ∞ ( R , U ) {\displaystyle c\in C^{\infty }(\mathbb {R} ,U)} . See [KM], 3.11.

## Main properties of smooth calculus

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1\. For maps on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions. On  R 2 {\displaystyle \mathbb {R} ^{2}} this is a non-trivial theorem, proved by Boman, 1967. See also [KM], 3.4.

2\. Multilinear mappings are smooth if and only if they are bounded ([KM], 5.5).

3\. If  f : E ⊇ U -> F {\displaystyle f:E\supseteq U\to F} is smooth then the derivative  d f : U × E -> F {\displaystyle df:U\times E\to F} is smooth, and also  d f : U -> L ( E , F ) {\displaystyle df:U\to L(E,F)} is smooth where  L ( E , F ) {\displaystyle L(E,F)} denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets; see [KM], 3.18.

4\. The chain rule holds ([KM], 3.18).

5\. The space  C ∞ ( U , F ) {\displaystyle C^{\infty }(U,F)} of all smooth mappings  U -> F {\displaystyle U\to F} is again a convenient vector space where the structure is given by the following injection, where  C ∞ ( R , R ) {\displaystyle C^{\infty }(\mathbb {R} ,\mathbb {R} )} carries the topology of compact convergence in each derivative separately; see [KM], 3.11 and 3.7.

C ∞ ( U , F ) -> ∏ c ∈ C ∞ ( R , U ) , ℓ ∈ F ∗ C ∞ ( R , R ) , f ↦ ( ℓ ∘ f ∘ c ) c , ℓ . {\displaystyle C^{\infty }(U,F)\to \prod _{c\in C^{\infty }(\mathbb {R} ,U),\ell \in F^{*}}C^{\infty }(\mathbb {R} ,\mathbb {R} ),\quad f\mapsto (\ell \circ f\circ c)_{c,\ell }\,.}

6\. The exponential law holds ([KM], 3.12): For  c ∞ {\displaystyle c^{\infty }} -open  V ⊆ F {\displaystyle V\subseteq F} the following mapping is a linear diffeomorphism of convenient vector spaces.

C ∞ ( U , C ∞ ( V , G ) ) ≅ C ∞ ( U × V , G ) , f ↦ g , f ( u ) ( v ) = g ( u , v ) . {\displaystyle C^{\infty }(U,C^{\infty }(V,G))\cong C^{\infty }(U\times V,G),\qquad f\mapsto g,\qquad f(u)(v)=g(u,v).}

This is the main assumption of variational calculus. Here it is a theorem. This property is the source of the name convenient, which was borrowed from (Steenrod 1967).

7\. Smooth uniform boundedness theorem ([KM], theorem 5.26). A linear mapping  f : E -> C ∞ ( V , G ) {\displaystyle f:E\to C^{\infty }(V,G)} is smooth (by (2) equivalent to bounded) if and only if  ev v ∘ f : V -> G {\displaystyle \operatorname {ev} _{v}\circ f:V\to G} is smooth for each  v ∈ V {\displaystyle v\in V} .

8\. The following canonical mappings are smooth. This follows from the exponential law by simple categorical reasonings, see [KM], 3.13.

ev : C ∞ ( E , F ) × E -> F , ev ( f , x ) = f ( x ) ins : E -> C ∞ ( F , E × F ) , ins ( x ) ( y ) = ( x , y ) ( ) ∧ : C ∞ ( E , C ∞ ( F , G ) ) -> C ∞ ( E × F , G ) ( ) ∨ : C ∞ ( E × F , G ) -> C ∞ ( E , C ∞ ( F , G ) ) comp : C ∞ ( F , G ) × C ∞ ( E , F ) -> C ∞ ( E , G ) C ∞ ( , ) : C ∞ ( F , F 1 ) × C ∞ ( E 1 , E ) -> C ∞ ( C ∞ ( E , F ) , C ∞ ( E 1 , F 1 ) ) , ( f , g ) ↦ ( h ↦ f ∘ h ∘ g ) ∏ : ∏ C ∞ ( E i , F i ) -> C ∞ ( ∏ E i , ∏ F i ) {\displaystyle {\begin{aligned}&\operatorname {ev} :C^{\infty }(E,F)\times E\to F,\quad {\text{ev}}(f,x)=f(x)\\\\[6pt]&\operatorname {ins} :E\to C^{\infty }(F,E\times F),\quad {\text{ins}}(x)(y)=(x,y)\\\\[6pt]&(\quad )^{\wedge }:C^{\infty }(E,C^{\infty }(F,G))\to C^{\infty }(E\times F,G)\\\\[6pt]&(\quad )^{\vee }:C^{\infty }(E\times F,G)\to C^{\infty }(E,C^{\infty }(F,G))\\\\[6pt]&\operatorname {comp} :C^{\infty }(F,G)\times C^{\infty }(E,F)\to C^{\infty }(E,G)\\\\[6pt]&C^{\infty }(\quad ,\quad ):C^{\infty }(F,F_{1})\times C^{\infty }(E_{1},E)\to C^{\infty }(C^{\infty }(E,F),C^{\infty }(E_{1},F_{1})),\quad (f,g)\mapsto (h\mapsto f\circ h\circ g)\\\\[6pt]&\prod :\prod C^{\infty }(E_{i},F_{i})\to C^{\infty }\left(\prod E_{i},\prod F_{i}\right)\end{aligned}}}

## Related convenient calculi

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Convenient calculus of smooth mappings appeared for the first time in [Frölicher, 1981], [Kriegl 1982, 1983]. Convenient calculus (having properties 6 and 7) exists also for:

* Real analytic mappings (Kriegl, Michor, 1990; see also [KM], chapter II).
* Holomorphic mappings (Kriegl, Nel, 1985; see also [KM], chapter II). The notion of holomorphy is that of [Fantappié, 1930-33].
* Many classes of Denjoy Carleman ultradifferentiable functions, both of Beurling type and of Roumieu-type [Kriegl, Michor, Rainer, 2009, 2011, 2015].
* With some adaptations,  Lip k {\displaystyle \operatorname {Lip} ^{k}} , [FK].
* With more adaptations, even  C k , α {\displaystyle C^{k,\alpha }} (i.e., the  k {\displaystyle k} -th derivative is Hölder-continuous with index  α {\displaystyle \alpha } ) ([Faure, 1989], [Faure, These Geneve, 1991]).

The corresponding notion of convenient vector space is the same (for their underlying real vector space in the complex case) for all these theories.

## Application: Manifolds of mappings between finite dimensional manifolds

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The exponential law 6 of convenient calculus allows for very simple proofs of the basic facts about manifolds of mappings. Let  M {\displaystyle M} and  N {\displaystyle N} be finite dimensional smooth manifolds where  M {\displaystyle M} is compact. We use an auxiliary Riemann metric g ¯ {\displaystyle {\bar {g}}} on  N {\displaystyle N} . The Riemannian exponential mapping of  g ¯ {\displaystyle {\bar {g}}} is described in the following diagram:

It induces an atlas of charts on the space  C ∞ ( M , N ) {\displaystyle C^{\infty }(M,N)} of all smooth mappings  M -> N {\displaystyle M\to N} as follows. A chart centered at  f ∈ C ∞ ( M , N ) {\displaystyle f\in C^{\infty }(M,N)} , is:

u f : C ∞ ( M , N ) ⊃ U f = { g : ( f , g ) ( M ) ⊂ V N × N } -> U ~ f ⊂ Γ ( f ∗ T N ) , {\displaystyle u_{f}:C^{\infty }(M,N)\supset U_{f}=\\{g:(f,g)(M)\subset V^{N\times N}\\}\to {\tilde {U}}_{f}\subset \Gamma (f^{*}TN),}
u f ( g ) = ( π N , exp g ¯ ) − 1 ∘ ( f , g ) , u f ( g ) ( x ) = ( exp f ( x ) g ¯ ) − 1 ( g ( x ) ) , {\displaystyle u_{f}(g)=(\pi _{N},\exp ^{\bar {g}})^{-1}\circ (f,g),\quad u_{f}(g)(x)=(\exp _{f(x)}^{\bar {g}})^{-1}(g(x)),}
( u f ) − 1 ( s ) = exp f g ¯ ∘ s , ( u f ) − 1 ( s ) ( x ) = exp f ( x ) g ¯ ⁡ ( s ( x ) ) . {\displaystyle (u_{f})^{-1}(s)=\exp _{f}^{\bar {g}}\circ s,\qquad \quad (u_{f})^{-1}(s)(x)=\exp _{f(x)}^{\bar {g}}(s(x)).}

Now the basics facts follow in easily. Trivializing the pull back vector bundle  f ∗ T N {\displaystyle f^{*}TN} and applying the exponential law 6 leads to the diffeomorphism

C ∞ ( R , Γ ( M ; f ∗ T N ) ) = Γ ( R × M ; p r 2 ∗ ⁡ f ∗ T N ) . {\displaystyle C^{\infty }(\mathbb {R} ,\Gamma (M;f^{*}TN))=\Gamma (\mathbb {R} \times M;\operatorname {pr_{2}} ^{*}f^{*}TN).}

All chart change mappings are smooth ( C ∞ {\displaystyle C^{\infty }} ) since they map smooth curves to smooth curves:

U ~ f 1 ∋ s ↦ ( π N , exp g ¯ ) ∘ s ↦ ( π N , exp g ¯ ) ∘ ( f 2 , exp f 1 g ¯ ∘ s ) . {\displaystyle {\tilde {U}}_{f_{1}}\ni s\mapsto (\pi _{N},\exp ^{\bar {g}})\circ s\mapsto (\pi _{N},\exp ^{\bar {g}})\circ (f_{2},\exp _{f_{1}}^{\bar {g}}\circ s).}

Thus  C ∞ ( M , N ) {\displaystyle C^{\infty }(M,N)} is a smooth manifold modeled on Fréchet spaces. The space of all smooth curves in this manifold is given by

C ∞ ( R , C ∞ ( M , N ) ) ≅ C ∞ ( R × M , N ) . {\displaystyle C^{\infty }(\mathbb {R} ,C^{\infty }(M,N))\cong C^{\infty }(\mathbb {R} \times M,N).}

Since it visibly maps smooth curves to smooth curves, composition

C ∞ ( P , M ) × C ∞ ( M , N ) -> C ∞ ( P , N ) , ( f , g ) ↦ g ∘ f , {\displaystyle C^{\infty }(P,M)\times C^{\infty }(M,N)\to C^{\infty }(P,N),\qquad (f,g)\mapsto g\circ f,}

is smooth. As a consequence of the chart structure, the tangent bundle of the manifold of mappings is given by

π C ∞ ( M , N ) = C ∞ ( M , π N ) : T C ∞ ( M , N ) = C ∞ ( M , T N ) -> C ∞ ( M , N ) . {\displaystyle \pi _{C^{\infty }(M,N)}=C^{\infty }(M,\pi _{N}):TC^{\infty }(M,N)=C^{\infty }(M,TN)\to C^{\infty }(M,N).}

### Regular Lie groups

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Let  G {\displaystyle G} be a connected smooth Lie group modeled on convenient vector spaces, with Lie algebra  g = T e G {\displaystyle {\mathfrak {g}}=T_{e}G} . Multiplication and inversion are denoted by:

μ : G × G -> G , μ ( x , y ) = x . y = μ x ( y ) = μ y ( x ) , ν : G -> G , ν ( x ) = x − 1 . {\displaystyle \mu :G\times G\to G,\quad \mu (x,y)=x.y=\mu _{x}(y)=\mu ^{y}(x),\qquad \nu :G\to G,\nu (x)=x^{-1}.}

The notion of a regular Lie group is originally due to Omori et al. for Fréchet Lie groups, was weakened and made more transparent by J. Milnor, and was then carried over to convenient Lie groups; see [KM], 38.4.

A Lie group  G {\displaystyle G} is called regular if the following two conditions hold:

* For each smooth curve  X ∈ C ∞ ( R , g ) {\displaystyle X\in C^{\infty }(\mathbb {R} ,{\mathfrak {g}})} in the Lie algebra there exists a smooth curve  g ∈ C ∞ ( R , G ) {\displaystyle g\in C^{\infty }(\mathbb {R} ,G)} in the Lie group whose right logarithmic derivative is  X {\displaystyle X} . It turn out that  g {\displaystyle g} is uniquely determined by its initial value  g ( 0 ) {\displaystyle g(0)} , if it exists. That is,

g ( 0 ) = e , ∂ t g ( t ) = T e ( μ g ( t ) ) X ( t ) = X ( t ) . g ( t ) . {\displaystyle g(0)=e,\qquad \partial _{t}g(t)=T_{e}(\mu ^{g(t)})X(t)=X(t).g(t).}

If  g {\displaystyle g} is the unique solution for the curve  X {\displaystyle X} required above, we denote

evol G r ⁡ ( X ) = g ( 1 ) , Evol G r ⁡ ( X ) ( t ) := g ( t ) = evol G r ⁡ ( t X ) . {\displaystyle \operatorname {evol} _{G}^{r}(X)=g(1),\quad \operatorname {Evol} _{G}^{r}(X)(t):=g(t)=\operatorname {evol} _{G}^{r}(tX).}

* The following mapping is required to be smooth:

evol G r : C ∞ ( R , g ) -> G . {\displaystyle \operatorname {evol} _{G}^{r}:C^{\infty }(\mathbb {R} ,{\mathfrak {g}})\to G.}

If  X {\displaystyle X} is a constant curve in the Lie algebra, then  evol G r ⁡ ( X ) = exp G ⁡ ( X ) {\displaystyle \operatorname {evol} _{G}^{r}(X)=\exp ^{G}(X)} is the group exponential mapping.

Theorem. For each compact manifold  M {\displaystyle M} , the diffeomorphism group  Diff ⁡ ( M ) {\displaystyle \operatorname {Diff} (M)} is a regular Lie group. Its Lie algebra is the space  X ( M ) {\displaystyle {\mathfrak {X}}(M)} of all smooth vector fields on  M {\displaystyle M} , with the negative of the usual bracket as Lie bracket.

Proof: The diffeomorphism group  Diff ⁡ ( M ) {\displaystyle \operatorname {Diff} (M)} is a smooth manifold since it is an open subset in  C ∞ ( M , M ) {\displaystyle C^{\infty }(M,M)} . Composition is smooth by restriction. Inversion is smooth: If  t -> f ( t , ) {\displaystyle t\to f(t,\ )} is a smooth curve in  Diff ⁡ ( M ) {\displaystyle \operatorname {Diff} (M)} , then f(t, )−1
f ( t , ) − 1 ( x ) {\displaystyle f(t,\ )^{-1}(x)} satisfies the implicit equation  f ( t , f ( t , ) − 1 ( x ) ) = x {\displaystyle f(t,f(t,\quad )^{-1}(x))=x} , so by the finite dimensional implicit function theorem,  ( t , x ) ↦ f ( t , ) − 1 ( x ) {\displaystyle (t,x)\mapsto f(t,\ )^{-1}(x)} is smooth. So inversion maps smooth curves to smooth curves, and thus inversion is smooth. Let  X ( t , x ) {\displaystyle X(t,x)} be a time dependent vector field on  M {\displaystyle M} (in  C ∞ ( R , X ( M ) ) {\displaystyle C^{\infty }(\mathbb {R} ,{\mathfrak {X}}(M))} ). Then the flow operator  Fl {\displaystyle \operatorname {Fl} } of the corresponding autonomous vector field  ∂ t × X {\displaystyle \partial _{t}\times X} on  R × M {\displaystyle \mathbb {R} \times M} induces the evolution operator via

Fl s ⁡ ( t , x ) = ( t + s , Evol ⁡ ( X ) ( t , x ) ) {\displaystyle \operatorname {Fl} _{s}(t,x)=(t+s,\operatorname {Evol} (X)(t,x))}

which satisfies the ordinary differential equation

∂ t Evol ⁡ ( X ) ( t , x ) = X ( t , Evol ⁡ ( X ) ( t , x ) ) . {\displaystyle \partial _{t}\operatorname {Evol} (X)(t,x)=X(t,\operatorname {Evol} (X)(t,x)).}

Given a smooth curve in the Lie algebra,  X ( s , t , x ) ∈ C ∞ ( R 2 , X ( M ) ) {\displaystyle X(s,t,x)\in C^{\infty }(\mathbb {R} ^{2},{\mathfrak {X}}(M))} , then the solution of the ordinary differential equation depends smoothly also on the further variable  s {\displaystyle s} , thus  evol Diff ⁡ ( M ) r {\displaystyle \operatorname {evol} _{\operatorname {Diff} (M)}^{r}} maps smooth curves of time dependent vector fields to smooth curves of diffeomorphism. QED.

### The principal bundle of embeddings

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For finite dimensional manifolds  M {\displaystyle M} and  N {\displaystyle N} with  M {\displaystyle M} compact, the space  Emb ⁡ ( M , N ) {\displaystyle \operatorname {Emb} (M,N)} of all smooth embeddings of  M {\displaystyle M} into  N {\displaystyle N} , is open in  C ∞ ( M , N ) {\displaystyle C^{\infty }(M,N)} , so it is a smooth manifold. The diffeomorphism group  Diff ⁡ ( M ) {\displaystyle \operatorname {Diff} (M)} acts freely and smoothly from the right on  Emb ⁡ ( M , N ) {\displaystyle \operatorname {Emb} (M,N)} .

Theorem: Emb ⁡ ( M , N ) -> Emb ⁡ ( M , N ) / Diff ⁡ ( M ) {\displaystyle \operatorname {Emb} (M,N)\to \operatorname {Emb} (M,N)/\operatorname {Diff} (M)} is a principal fiber bundle with structure group  Diff ⁡ ( M ) {\displaystyle \operatorname {Diff} (M)} .

Proof: One uses again an auxiliary Riemannian metric  g ¯ {\displaystyle {\bar {g}}} on  N {\displaystyle N} . Given  f ∈ Emb ⁡ ( M , N ) {\displaystyle f\in \operatorname {Emb} (M,N)} , view  f ( M ) {\displaystyle f(M)} as a submanifold of  N {\displaystyle N} , and split the restriction of the tangent bundle  T N {\displaystyle TN} to  f ( M ) {\displaystyle f(M)} into the subbundle normal to  f ( M ) {\displaystyle f(M)} and tangential to  f ( M ) {\displaystyle f(M)} as  T N | f ( M ) = Nor ⁡ ( f ( M ) ) ⊕ T f ( M ) {\displaystyle TN|_{f(M)}=\operatorname {Nor} (f(M))\oplus Tf(M)} . Choose a tubular neighborhood

p f ( M ) : Nor ⁡ ( f ( M ) ) ⊃ W f ( M ) -> f ( M ) . {\displaystyle p_{f(M)}:\operatorname {Nor} (f(M))\supset W_{f(M)}\to f(M).}

If  g : M -> N {\displaystyle g:M\to N} is  C 1 {\displaystyle C^{1}} -near to  f {\displaystyle f} , then

ϕ ( g ) := f − 1 ∘ p f ( M ) ∘ g ∈ Diff ⁡ ( M ) and g ∘ ϕ ( g ) − 1 ∈ Γ ( f ∗ W f ( M ) ) ⊂ Γ ( f ∗ Nor ⁡ ( f ( M ) ) ) . {\displaystyle \phi (g):=f^{-1}\circ \,p_{f(M)}\circ \,g\in \operatorname {Diff} (M)\quad {\text{and}}\quad g\circ \,\phi (g)^{-1}\in \Gamma (f^{*}W_{f(M)})\subset \Gamma (f^{*}\operatorname {Nor} (f(M))).}

This is the required local splitting. QED

### Further applications

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An overview of applications using geometry of shape spaces and diffeomorphism groups can be found in [Bauer, Bruveris, Michor, 2014].

## Notes

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1. ^ An example of a composition mapping is the evaluation mapping  ev : E × E ∗ -> R {\displaystyle {\text{ev}}:E\times E^{*}\to \mathbb {R} } , where  E {\displaystyle E} is a locally convex vector space, and where  E ∗ {\displaystyle E^{*}} is its dual of continuous linear functionals equipped with any locally convex topology such that the evaluation mapping is separately continuous. If the evaluation is assumed to be jointly continuous, then there are neighborhoods  U ⊆ E {\displaystyle U\subseteq E} and  V ⊆ E ∗ {\displaystyle V\subseteq E^{*}} of zero such that  U × V ⊆ [ 0 , 1 ] {\displaystyle U\times V\subseteq [0,1]} . However, this means that  U {\displaystyle U} is contained in the polar of the open set  V {\displaystyle V} ; so it is bounded in  E {\displaystyle E} . Thus  E {\displaystyle E} admits a bounded neighborhood of zero, and is thus a normed vector space.
2. ^ In order to be useful for solving equations like nonlinear PDE's, convenient calculus has to be supplemented by, for example, a priori estimates which help to create enough Banach space situation to allow convergence of some iteration procedure; for example, see the Nash–Moser theorem, described in terms of convenient calculus in [KM], section 51.

## References

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* Bauer, M., Bruveris, M., Michor, P.W.: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. (arXiv:1305.11500)
* Boman, J.: Differentiability of a function and of its composition with a function of one variable, Mathematica Scandinavia vol. 20 (1967), 249–268.
* Faure, C.-A.: Sur un théorème de Boman, C. R. Acad. Sci., Paris}, vol. 309 (1989), 1003–1006.
* Faure, C.-A.: Théorie de la différentiation dans les espaces convenables, These, Université de Genève, 1991.
* Frölicher, A.: Applications lisses entre espaces et variétés de Fréchet, C. R. Acad. Sci. Paris, vol. 293 (1981), 125–127.
* [FK] Frölicher, A., Kriegl, A.: Linear spaces and differentiation theory. Pure and Applied Mathematics, J. Wiley, Chichester, 1988.
* Kriegl, A.: Die richtigen Räume für Analysis im Unendlich – Dimensionalen, Monatshefte für Mathematik vol. 94 (1982) 109–124.
* Kriegl, A.: Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektorräumen, Monatshefte für Mathematik vol. 95 (1983) 287–309.
* [KM] Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997. (pdf)
* Kriegl, A., Michor, P. W., Rainer, A.: The convenient setting for non-quasianalytic Denjoy–Carleman differentiable mappings, Journal of Functional Analysis, vol. 256 (2009), 3510–3544. (arXiv:0804.2995)
* Kriegl, A., Michor, P. W., Rainer, A.: The convenient setting for quasianalytic Denjoy–Carleman differentiable mappings, Journal of Functional Analysis, vol. 261 (2011), 1799–1834. (arXiv:0909.5632)
* Kriegl, A., Michor, P. W., Rainer, A.: The convenient setting for Denjoy-Carleman differentiable mappings of Beurling and Roumieu type. Revista Matemática Complutense (2015). doi:10.1007/s13163-014-0167-1. (arXiv:1111.1819)
* Michor, P.W.: Manifolds of mappings and shapes. (arXiv:1505.02359)
* Steenrod, N. E.: A convenient category for topological spaces, Michigan Mathematical Journal, vol. 14 (1967), 133–152.

* v
* t
* e

Analysis in topological vector spaces

Basic concepts|

* Abstract Wiener space
* Classical Wiener space
* Bochner space
* Convex series
* Cylinder set measure
* Infinite-dimensional vector function
* Matrix calculus
* Vector calculus

Derivatives|

* Differentiable vector–valued functions from Euclidean space
* Differentiation in Fréchet spaces
* Fréchet derivative
* Total
* Functional derivative
* Gateaux derivative
* Directional
* Generalizations of the derivative
* Hadamard derivative
* Holomorphic
* Quasi-derivative

Measurability|

* Besov measure
* Cylinder set measure
* Canonical Gaussian
* Classical Wiener measure
* Measure like set functions
* infinite-dimensional Gaussian measure
* Projection-valued
* Vector
* Bochner / Weakly / Strongly measurable function
* Radonifying function

Integrals|

* Bochner
* Direct integral
* Dunford
* Gelfand–Pettis/Weak
* Regulated
* Paley–Wiener

Results|

* Cameron–Martin theorem
* Inverse function theorem
* Nash–Moser theorem
* Feldman–Hájek theorem
* No infinite-dimensional Lebesgue measure
* Sazonov's theorem
* Structure theorem for Gaussian measures

Related|

* Crinkled arc
* Covariance operator

Functional calculus|

* Borel functional calculus
* Continuous functional calculus
* Holomorphic functional calculus

Applications|

* Banach manifold (bundle)
* Convenient vector space
* Choquet theory
* Fréchet manifold
* Hilbert manifold

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Topological vector spaces (TVSs)

Basic concepts|

* Banach space
* Completeness
* Continuous linear operator
* Linear functional
* Fréchet space
* Linear map
* Locally convex space
* Metrizability
* Operator topologies
* Topological vector space
* Vector space

Main results|

* Anderson–Kadec
* Banach–Alaoglu
* Closed graph theorem
* F. Riesz's
* Hahn–Banach (hyperplane separation
* Vector-valued Hahn–Banach)
* Open mapping (Banach–Schauder)
* Bounded inverse
* Uniform boundedness (Banach–Steinhaus)

Maps|

* Bilinear operator
* form
* Linear map
* Almost open
* Bounded
* Continuous
* Closed
* Compact
* Densely defined
* Discontinuous
* Topological homomorphism
* Functional
* Linear
* Bilinear
* Sesquilinear
* Norm
* Seminorm
* Sublinear function
* Transpose

Types of sets|

* Absolutely convex/disk
* Absorbing/Radial
* Affine
* Balanced/Circled
* Banach disks
* Bounding points
* Bounded
* Complemented subspace
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Extreme point
* Pre-compact/Totally bounded
* Prevalent/Shy
* 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
* BK-space
* (Ultra-) Bornological
* Brauner
* Complete
* Convenient
* (DF)-space
* Distinguished
* F-space
* FK-AK space
* FK-space
* Fréchet
* tame Fréchet
* Grothendieck
* Hilbert
* Infrabarreled
* Interpolation space
* K-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

* Category

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Functional analysis (topics – glossary)

Spaces|

* Banach
* Besov
* Fréchet
* Hilbert
* Hölder
* Nuclear
* Orlicz
* Schwartz
* Sobolev
* Topological vector

Properties|

* Barrelled
* Complete
* Dual (Algebraic / Topological)
* Locally convex
* Reflexive
* Separable

Theorems|

* Hahn–Banach
* Riesz representation
* Closed graph
* Uniform boundedness principle
* Kakutani fixed-point
* Krein–Milman
* Min–max
* Gelfand–Naimark
* Banach–Alaoglu

Operators|

* Adjoint
* Bounded
* Compact
* Hilbert–Schmidt
* Normal
* Nuclear
* Trace class
* Transpose
* Unbounded
* Unitary

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|

* Hardy space
* Spectral theory of ordinary differential equations
* Heat kernel
* Index theorem
* Calculus of variations
* Functional calculus
* Integral linear operator
* Jones polynomial
* Topological quantum field theory
* Noncommutative geometry
* Riemann hypothesis
* Distribution (or Generalized functions)

Advanced topics|

* Approximation property
* Balanced set
* Choquet theory
* Weak topology
* Banach–Mazur distance
* Tomita–Takesaki theory

* Category

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