Local Smooth Functions
by Heinrich Hartmann / 2023-06-08 / Boltenhagen / source
Abstract
In this note, we develop the classical analysis in the context of "local" \(C^\infty\) functions in \(n\) variables.
Local functions are functions that are only defined on the neighborhood of the origing, and equalities between local functions are only required to hold after possibly shrinking the neighbourhood further.
In this setting, statements can be formulated more concisely, as we don't need to include choices of \(\eps\) at every step. One example is the theorem of inverse functions, which states that \(\phi\) is invertible as local function if and only if \(D_0 \phi\) is invertible as linear map (i.e. \(n \times n\) matrix).
Local functions are frequently studied the in context of commutative algebra (cf. localisation), i.e. when studying polynomial functions or integres. We prove that local \(C^\infty\) functions are indeed a localisation in the sense of commutative algebra. Somewhat surprisingly, they are at the same time a quotient ring of the ring of \(C^\infty\) functions. The Taylor series give us a linar map from local functions to another classical commutative algebra concept, the ring of formal power series. A highly surprising theorem of Borel shows, this map is surjective, i.e. all formal power series can be realized as taylor series of smooth local functions.
This is a living document that may be expanded in the future. At the time of this writing, the study introduces consisten notations for (differential) operators on local functions, and coveres Taylor Series, Hadamard Lemma, Tangent and Cotangent spaces and the Theorem of implicit functions.
Local \(C^\infty\) functions
Definition. The local ring of \(C^\infty\) functions around \(0 \in \IR^n\) is defined as
The inductive limit is taken along the restriction maps \(\rho_V: \CINF(U,\IR) \ra \CINF(V,\IR)\) for \(U \subset V\).
This means elements \(a \in \AINF\) are represented by a pairs \((U,a)\), where \(U \subset \IR^n\) is an open subset containing \(0\), and \(a \in \AINF(U,\IR)\). Two pairs \((U,a), (V,b)\) represent the same element in \(\AINF_n\) if there is a \(W \subset U \cap V, 0 \in W\) so that \(\rho_W(a) = \rho_W(b)\).
Naturality. Any (\(C^\infty\)-)differentiable map \(f: \IR^n \ra \IR^m\) with \(f(0) = 0\) gives rise to a map \(f^*: \AINF_m \ra \AINF_n\), which sends \((b,V)\) to \((b \circ f, f^{-1}(V))\). We call \(f^*[a]\) the pullback of a along \(f\).
Evaluation. For \(a \in \AINF_n\), we denote by \(e[a] := ev_0[a] := a(0) \in \IR\), the evaluation of \(a\) at \(0 \in \IR^n\). This is operation can be identified with the pullback along the injection \(0: \IR^0 \ra \IR^n\), \(e = 0^*\).
Constants. For \(\lambda \in \IR\) the constant function \(\IR^n \ra \IR, x \mapsto \lambda\) by \(\underline{\lambda} \in \AINF_n\) or simply \(\lambda\) when there is no risk of confusion.
The linear map \(\IR \ra \AINF_n, \lambda \mapsto \underline{\lambda}\) can be identified with \(\pi^*\), where \(\pi:\IR^n \ra \IR^0\) is the projection to a point.
Elementary Operations. Let \(i \in \{1,\dots,n\}\). We have the following operations on \(\AINF_n\).
- Coordinate fuctions \(x^i \in \AINF_n\) with \(x^i(x_1,\dots,x_n) = x_i\) on any \(0 \in U \subset \IR^n\),
- Partial derivatives \(\del_i = \del_i^x = \del_{x^i}\) those are linear operators \(\AINF_n \ra \AINF_n\).
- Axis injections \(x_i: \IR \ra \IR^n\), giving rise to maps \(x_i^*: \AINF_n \ra \AINF^1\) via pullback.
For a multi-index \(I \in \IN^n, I = (i_1, \dots, i_n)\), we define:
- The monomial \(x^I = (x^1)^{i_1} \cdot \dots \cdot (x^n)^{i_n} \in \AINF_n\)
- The differential operator \(\del^I = \del_1^{i_1} \cdot \dots \cdot \del_n^{i_n}\)
Elementary Relations.
- Product rule. \(\del_i[ a \cdot b ] = \del_i[a] \cdot b + a \cdot \del_i[b]\).
-
Iterated Product Rule.
\[ \del^I[a \cdot b] = \sum_{J,K \in \IN^n, J+K=I} \frac{I!}{J! K!} \del^J[a] \cdot \del^K[b]. \]
-
Chain Rule.
\[ \del_i [f^*[b]] = \del_i[ b \circ f ] = \sum_{k=1,\dots,m} \del_k[b] \circ f \cdot \del_i[f^k] = \sum_{k=1,\dots,m} f^*[ \del_k[b] ] \cdot \del_i[f^k], \]the components \(f^k = x^k \circ f\).
Taylor Series
Definition. The taylor series of \(a \in \AINF_n\) is the formal power series
where \(I! = i_1! \cdot \dots \cdot i_n!\). The degree-k Taylor polyonmial is the degree-k truncation of the taylor series:
Properties.
-
For a polynomial \(p \in \IR[x^1, \dots, x^n]\), we have \(T_k[p] = p\) if \(k \geq deg(p)\).
-
For \(a,b \in \AINF_n\), we have
$$ T(a \cdot b) = T(a) \cdot T(b), \quad\text{and}\quad T_k(a) \cdot T_k(b). $$
-
Let \(A: \IR^m \ra \IR^n\) be a linear map and \(a \in \AINF_n\), then
$$ % need spaces here to avoid Mardown parsing as links T[a \circ A] (x) = T[a] (A \cdot x). $$
-
In particular, for a fixed \(x \in \IR^n\) the Taylor series of the 1-dimensional function \([t \mapsto a(x \cdot t)]\in \AINF^1\) is the restriction of the full Taylor series of \(a\):
$$ T[t \mapsto a(x \cdot t)] (t) = T[a] (x \cdot t). $$
Proof
All three statements are proved via direct comuptation. Statement (2) follows from the iterated product rule stated above.
To prove (3) one verifies that both sides are equal to: $$ \sum_{M \in \IN^{n \times m}, \sigma_1(M)=J, \sigma_2(M)=I} \del^I [a] (0) \cdot \frac{A^M}{M!} \cdot y^J $$
Here the sum runs over all two-dimensional multi-indices \(M \in \IN^{n \times m}\). And we denote by \(\sigma_i\) the summation of the \(i\)-th index: \(\sigma_1(M) = \sum_i M_{i,j} \in \IN^m\), \(\sigma_2(M)=\sum_j M_{i,j} \in \IN^n\). Furthermore \(A^M = \prod_{i,j}A_{i,j}^{M_{i,j}}\) and \(M! = \prod_{i,j} M_{i,j}!\).
Proposition(Functoriality/Faà di Bruno)
Let \(f:U \subset \IR^m \ra \IR^n\) be a function defined in a neighbourhood of he origin \(0 \in \IR^m\). Let \(f^1,\dots,f^n\) be the components of \(f\). We define \(T[f]\) as the element \((T[f^1], \dots, T[f^n]) \in \IR \dbrackets{ x^1, \dots, x^m }^{\vsum n}\). Let \(a \in \AINF_n\), then:
As composition of formal power series. Comparing coefficients yields formulas for the derivaives of the composition. The general case is surprisingly very hard and has only been established in full generality in by Constantine and Savits in 1996 The special case \(n=m=1\) has been estabished by Faà di Bruno in 1855 (Faa di Bruno, C. F. (1855). Note sur une nouvelle formule du calcul differentiel, Quart. J. Math.)
Let \(\Pi_k\) be the set of all partitions \(\pi\) of \(\{1,\dots,k\}\). \(|\pi|\) is the number of blocks of \(\pi\). For each block \(B\in\pi\) let \(|B|\) be is its size. Then:
Proof
For the combinatorics of how to derive the Faa di Bruno formula, please consult the literature.
For proving the naturality of \(T\): \(T[a \circ f] = T[a] \circ T[f]\), there are two broad strategies:
- Exploit the product structure to reduce to linear maps which had been established.
- Use the functoriality of completion / direct limits.
The second approach uses concepts developed in the Commutative Algebra section below, and exploits that \(T\) induces an isomorphism between the \(\fm\)-adic completion of \(\AINF_n\) and the formal power series ring \(\IR \dbrackets{ x^1, \dots, x^m }\). As \(\fm\)-adic completion is a functorial construction the naturality of \(T\) follows.
Theorem (Residuals) For any \(a \in \AINF\) on any convex set \(U\) of definition we set \(R_{k+1} = a - T_k[a]\).
-
(Qualitative Residual). For all \(I \in \IN^n\) with \(|I| \leq k\) we have
$$ \del^I [R_{k+1}] (0) = 0. $$
-
(Integral Residual)
$$ R_{k+1}(x) = (k+1) \sum_{|I| = k+1} \frac{x^I}{I!} \int_0^1 \del^I[a] (t \cdot x) \cdot (1-t)^k dt. $$
-
(Lagrange Residual) For all \(x \in U\) there exists a \(t \in [0,1]\) so that
$$ R_{k+1}(x) = \sum_{|I| = k+1} \frac{x^I}{I!} \del^I[a] (x \cdot t). $$
Proof
Property (1) follows from \(\del^I[x^J](0) = I! \cdot \delta_{I,J}\). For (2), (3) we reduce to the one-dimensional case using Property 4 above. Let \(a_x(t) = a(x \cdot t) \in \AINF^1\). We compute the derivatives to $$ \del [a_x] = \sum_i \del_i [a] (x\cdot t) x_i, \qtext{and} \del^n[a_x] = \sum_{|I|=n} \frac{n!}{I!} \cdot \del^I[a] (x \cdot t) \cdot x^I. $$
Then \(R_{k+1}(x) = a(x) - T_k [a] (x) = a_x(1) - T_k [a_x] (1)\). Now by the 1-dimensional result and iterated chain rule we have: $$ R_{k+1}(x) = \int_0^1 \del^{k+1} [a_x] (t) \frac{(1-t)^k}{k!} dt = \sum_{|I| = k+1 } \int_0^1 \del^{I} [a] (x \cdot t) \cdot x^I \cdot \frac{(k+1)!}{I!} \cdot \frac{(1-t)^k}{k!} dt. $$
Similarly, by the 1-d Lagrange residual formula, there exists a \(t \in [0,1]\) with $$ R_{k+1}(x) = \frac{ \del^{k+1} [a_x] (t) }{(k+1)!} = \sum_{|I| = k+1 } \del^{I} [a] (x \cdot t) \frac{x^I}{I!} dt. $$
Real Analytic Functions
Definition
We call a germ \(a \in \AINF_n\) real-analytic, if it's Taylor series \(T[a]\) converges and is equal to \(a\) in a neighbourhood of \(0\). We denote the set of real-analytic germs by \(\KA_n \subset \AINF_n\).
Proposition
The Taylor series induces and isomorphism \(T: \KA_n \ra \IR\{ x^1, \dots, x^n \}\) to the ring of convergent power series. \(\square\)
Theorem Let \(a \in \AINF_n\). The following are equivalent:
-
The germ \(a\) is real-analytic, \(a \in \KA_n\).
-
There is a representative of \(a\) in \(\CINF(U)\), so that for all compact subsets \(K \subset U\), there are constants \(C,R>0\) such that for all \(I \in \IN^n\):
Note that the estimates for \(K=\{0\}\) in (2) are sufficient to ensure convergence of the power series \(T[a]\). The bounds on compact neighbourhoods are necessary to ensure that the function represents \(a\) on an open subset.
Proof
(1) \(\Rightarrow\) (2): If \(a\) extends holomorphically to the complex polydisc \(P_r=\{z\in\IC^n:|z_j|<r\text{ for }1\le j\le n\}\), pick any \(0<r'<r\). Then for \(\|x\|_\infty\le r'\) and \(I\in\IN^n\), we have \(\del^I[a](x) = \frac{I!}{(2\pi i)^n} \int_{|z_j-x_j|=r-r'} a(z)(z_1-x_1)^{-i_1-1}\cdots(z_n-x_n)^{-i_n-1} dz_1\cdots dz_n\), so \(|\del^I[a](x)| \le \sup_{P_r}|a| \cdot (r-r')^{-|I|} \cdot I! = C R^{|I|} I!\), where \(C=\sup_{P_r}|a|\) and \(R=(r-r')^{-1}\). Thus \(a \in G^1\) on \(\|x\|_\infty\le r'\).
(2) \(\Rightarrow\) (1): Assume \(|\del^I[a](x)|\le C R^{|I|} I!\) for \(\|x\|_\infty\le\varepsilon\). To show that \(T[a]\) converges and equals \(a\) it suffices to bound the integral-remainder form of the multivariate Taylor expansion: \(a(x) - T_N[a] = (N+1) \sum_{|I|=N+1} x^I (I!)^{-1} \int_0^1(1-t)^N \del^I[a](tx) dt\). For \(\|x\|_\infty < r\), we can bound each term by: \((N+1) \sum_{|I|=N+1} r^{N+1} (I!)^{-1} r^N C R^{N} (I!) = C_1 (rR)^{N}\). which converges to 0 as \(N\ra \infty\) if \(r < 1/R\). Hence \(T[a](x)\) converges uniformly to \(a(x)\) on \(\|x\|_\infty < 1/R\) and we see that \(a\) is real-analytic.
Commutative Algebra
\(\IR\)-Algebra Structure. The ring \(\AINF_n\) inherits the structure of a unital \(\IR\) algebra from \(\CINF(U,\IR)\). In other words elements \(a,b \in \AINF_n\) can be added, multiplied and multiplied by scalars \(\lambda \in \IR\), so that the usual associativity and distributivity relations hold. The unit element \(1 \in \AINF_n\) is the constant function \(1\).
- The structure maps \(\rho_0: \CINF(U,\IR) \ra \AINF_n\) are morphisms of \(\IR\)-algebras.
- The pullback maps \(f^*\) are morphisms of \(\IR\)-algebras.
- The "constants map" \(\lambda \mapsto \underline{\lambda}\) is a morphism of \(\IR\)-algebras.
- The evaluation map \(ev_0: a \mapsto a(0)\) is a morphism of \(\IR\)-algebras.
Locality
- Denote the kernel of the evaluation map by \(\fm = \ker(e) = \Set{ a \in \AINF_n }{ a(0) = 0 } \subset \AINF_n\).
- It is a maximal ideal in \(\AINF_n\).
- There is only one maximal ideal in \(\AINF_n\).
- The pullback maps \(\vphi^*: \AINF_m \ra \AINF_n\) induced by \(\vphi: \IR^n \ra \IR^m\) are local, in the sense that \(\vphi^*(\fm_m) \subset \fm_n\) or equivalently \(\vphi^{-1}(\fm_n) = \fm_m\).
Proof
The ideal \(\fm \subset \AINF_n\) is maximal, since every element not in \(\fm\) is a unit.
Let \(\fn \subset \AINF_n\) be another maximal ideal. We have \(\fn \subset \fm\) since otherwise \(\fn\) would contain a unit. Since \(\fn\) is maximal it follows that \(\fn = \fm\).
If \(b(0)\), then \(\vphi^*[b](0) = (b \circ \vphi) (0) = 0\), since \(\vphi(0) = 0\). Conversely if \(\vphi^*[b](0) = 0\) then \(b(0) = b \circ \vphi(0) = \vphi^*b(0) = 0\).
-
The maxiamal \(\fm = \Set{a}{a(0) = 0}\) ideal is generated by the coordinate fuctions \(x^i\):
$$ \fm = (x^1, \dots, x^n) \subset \AINF_n. $$
In other words, every element \(a \in \fm\) with \(a(0) = 0\) can be written in the form \(a = \sum_{i=1}^n x^i \cdot a_i\) with \(a_i \in \AINF_n\).
-
The kernel \(I_k\) of the degree-k Taylor map \(T_k: \AINF_n \ra \IR[x^1, \dots, x^n]\), is generated by \(x^{I}\), with \(|I|=k+1\).
$$ \ker(T_k) = \fm^{k+1} \subset \AINF_n $$
In other word, every element \(a \in \fm\) with \(a(0) = 0\) and \(\del^I[a] (0) = 0\) for all \(|I| \leq k\), can be written in the form \(a = \sum_{|I| = k+1}^n x^I \cdot a_I\), with \(a_I \in \AINF_n\).
Proof
(1) is a special case of (2) for k=1, but we include a full proof of the classical result for completeness:
Let \(a \in \fm\), and set \(a_i(x) = \int_0^1 \del_i[a](t \cdot x) dt\). Now \(\frac{\del}{\del t} a(t \cdot x) = \sum_{i=1}^n \del_i[a](tx) \cdot x^i(x)\), hence
(2) If \(a = \sum_{|I| = k+1}^n x^I \cdot a_I\), then \(T_k[a] = \sum_{|I| = k+1}^n T_k[x^I] \cdot T_k[a_I]\), but \(T_k[x^I] = 0\) if \(|I|=k+1\).
If \(T_k[a] = 0\), then we have by the Integral Residual formula: $$ a = a - T_k[a] = (k+1) \sum_{|I| = k+1} \frac{x^I}{I!} \int_0^1 \del^I[a] (t \cdot x) \cdot (1-t)^k dt = \sum_{|I| = k+1} x^I a_I. $$ where \(a_I = \frac{k+1}{I!} \int_0^1 \del^I[a] (t \cdot x) \cdot (1-t)^k dt\)
Remark. In the proof we saw that the Hadamard Lemma trivially follows from the Residual formula for the Taylor series presented above.
Corollary.
- If \(a = \sum_{i=1}^n x^i \cdot a_i\) is a Hadamard representation of \(a \in \fm\), then \(\del_i[a] (0) = a_i(0)\). Hence the function \(a_i\) can be treated as an algebraic replacement for the derivative at \(0\) in some settings.
- The morphisms of \(\IR\) algebras given by inclusion and degree-k Taylor expansion $$ \IR[x^1, \dots, x^n] \overset{\iota}{\lra} \AINF_n \overset{T_k}{\lra} \IR[x^1, \dots, x^n]/(x^1, \dots, x^n)^{k+1}. $$ induce an inverse pair of isomorphisms when passing to the quotients $$ \IR[x^1, \dots, x^n] / (x^1, \dots, x^n)^{k+1} \overset{\sim}{\lra} \AINF_n / (x^1, \dots, x^n)^{k+1} \overset{\sim}{\lra} \IR[x^1, \dots, x^n]/(x^1, \dots, x^n)^{k+1}. $$
-
The intersection of all \(\fm^k\) consists of germs of "flat functions" where all derivatives vanish:
\[ \cap_{k \geq 0} \fm^k = \ker(T) = \{ a \,|\, \del^I[a] = 0 \stext{for all} I\}\]
- The completion \(\hat \AINF_n = \varprojlim_k \AINF_n/\fm^k\) of \(\AINF_n\) at \(\fm_0\) is isomorphic to \(\IR\dbrackets{ x^1, \dots, x^n}\) via the Taylor Map: $$ T: \hat \AINF_n \lra \IR\dbrackets{ x^1, \dots, x^n}. $$
Proposition
For an open set \(0 \in U \subset \IR^n\), denote the restriction map by \(\rho: C^\infty(U, \IR) \ra \AINF_n\), let \(\fm_0 \subset C^\infty(U, \IR)\) be the kernel of the evaluation map at \(0\). Then,
- \(\rho\) is surjective and induces and isomorphism \(C^\infty(U, \IR) / \ker(\rho) \isom \AINF_n\).
- \(\rho\) induces an isomophism \(C^\infty(U, \IR)_{\fm_0} \isom \AINF_n\).
Proof
To prove the first claim choose a representative \((V, a) \in \AINF_n\), with \(V \subset U\). Choose \(\eps > 0\) small enough so that the ball \(B_{2\eps}\) lies in \(V\). Choose a bump function \(\delta\) with \(\delta = 1\) on \(B_{\eps}\) and \(\delta = 0\) outside of \(B_{2\eps}\). Then \(a \cdot \delta\) is defined on \(V\) and equal to zero outside of \(B_{2\eps}\), and hence extends to U.
To prove the second claim we first note that \(b(0) \neq 0\) implies that \(b(x) \neq 0\) for \(x\) in an open subset of \(0\), hence \(b\) is invertible in \(\AINF_n\). This shows that \(\rho\) induces a map from the localization. The map is surjective by the claim we just proved. To show the map is injective, assume that the formal quotient \((a,b)\) maps to \(a/b = 0 \in \AINF_n\). This means that \(a = 0\) in an open neightbourhood \(V\) of \(0\). Let \(\delta\) be a bump function with \(\delta(1) = 1\) and \(\delta(x) = 0\) outside of \(V\). Then \(\delta \notin \fm_0\) but \(\delta a = 0 \in C^\infty(U, \IR)\), showing that \((a,b) = 0 \in C^\infty(U, \IR)_{\fm_0}\).
Theorem (Borel). For every power series \(p \in \IR\dbrackets{x^1, \dots, x^n}\) there exists a \(C^\infty\) function \(a \in \AINF_n\) with \(T(a) = p\).
In other words, the map \(T : \AINF_n \to \IR\dbrackets{x^1, \dots, x^n}\) is surjective, and hence \(\IR\dbrackets{x^1, \dots, x^n} \cong \AINF_n / \ker(T)\).
For a proof see Ieke Moerdijk, Gonzalo E. Reyes: Models for Smooth Infinitesimal Analysis, p13. or ncatlab. The basic idea is to compose \(a\) as an infinite linear combination of bump functions that are realizing higher taylor coefficients while ensuring convergence. The Whitney extension theorem discussed below generalizes this existence results to arbitrary closed subsets with prescribed "normal jets".
Flat Functions
Definition
Elements \(a \in \AINF_n\) where all derivatives at \(0\) vanish are called flat functions. Flat functions form an ideal which we donet by \(\fF\). As we have seen above the following identity holds:
We get an exact sequence:
Example
- \(\phi(x) = exp(-1/x^2) \in \fF\) is called the standard/even flat function. We have \(\phi(x) > 0\) and \(\phi(0) = 0\).
- \(\psi(x) = sign(x) \cdot exp(-1/x^2) \in \fF\) is called the odd flat function. \(\psi\) a differentiable homoemorphism around \(0\), but not a diffeomorphism.
- \(θ(x) = exp(-1/x^2) \sin(1/pi x) \in \fF\) is an oscillatory flat function with zeros at the points \(0\), \(1/n\), \(-1/n\) for \(n \geq 0\).
Proposition
- If \(a^n \in \fF\) then \(a \in \fF\), i.e. \(\fF\) is reduced.
- If \(a \cdot b \in \fF\) then \(a \in \fF\) of \(b \in \fF\), i.e. \(\fF\) is a prime ideal.
- If \(a_n \in \fF\) is a sequence defined on a fixed \(U\), and \(a_n \ra a\) in the \(C^k\)-subnorms for all \(k\), then \(a \in \fF\), i.e. \(\fF\) is closed.
- \(\fF^{\infty} = \cap_k \fF^k\) is non-empty, e.g. \(\phi \in \fF^{\infty}\), i.e. \(\phi\) is infinitely divisible.
- ??? \(\fF \supset \fF^2 \supset \fF^k \dots\) are strict inclusions with infinite dimensional quotients.
Note
- The first two properties follow from the fact that \(\FPOW\) is an integral domain and \(T\) is the quotient map \(0 \ra \fF \ra \AINF_n \ra \FPOW \ra 0\).
- The closedness of the ideal follows since \(T\) is continuous in the Frechet topology on \(\CINF(U)\) and the adic topology in \(\fF\).
- To show that \(\phi(x)\) is infinitely divisible note that \(\phi(x)^k = \exp(-k/x^2)\), so we can give a \(k\)-th root as \(\exp(-1/k x^2)\) for all \(k\).
Whitney Extension
Definition
Denote by \( C^\infty(U) \) the ring of smooth real-valued functions on \( U \).
-
For a closed subset \(E\subset U\), define the vanishing ideal to be
\[ \mathcal{I}(E) := \{ f \in C^\infty(U) \mid f|_E = 0 \}. \]
-
We define the higher vanishing ideals as:
- \(\mathcal{I}_k(E) := \{ f \in C^\infty(U) \mid \del^I f|_E = 0 \;\forall\; |I| \leq k \}.\)
- \(\mathcal{I}_\infty(E) := \cap_k \mathcal{I}_k(E).\)
Theorem(Whitney/Malgrange)
For all closed subsets \(E\) the following sequence is exact:
- The quotient admits an explicit description as Whitney-functions on \(E\).
- The exact admints a splitting \(W\) that is continuous.
XXX
-
For an ideal \(I \subset \CINF(U)\) we define the vanishing locus to be
\[ \mathcal{Z}(I) := \{ x \in U \mid \forall f \in I,\ f(x) = 0 \}. \]
- For \(I \subset C^\infty(U)\), define the \(\Lambda\)-radical: $$ \sqrt[\Lambda]{I} := \Set{ f \in C^\infty(U) }{ \exists \varphi \in \Lambda \ \text{s.t.} \ \varphi \circ f \in I }. $$ where \(\Lambda := \Set{\varphi \in C^\infty(\IR)}{\varphi^{-1}(0) = \{0\}}\). Examples include \(\varphi(t) = t^{n}\) and \(\phi(t) = e^{-1/t^2}\) with \(\phi(0) = 0\).
Theorem (Smooth Nullstellensatz \(\Lambda\)-Form)
For every ideal \(I \subset C^\infty(U)\) and closed subset \(E \subset U\) we have.
In words:
- Every closed subset (in the Euklidean Topology) can be recovered as zero set of \(\CINF\) functions on \(U\).
- Every smooth function vanishing on \(E=Z(I)\) can be written as the sum of an element from the \(\Lambda\)-radical of \(I\) and a function flat along \(E\).
Proof
\(\mathcal{Z}(\mathcal{I}(E)) = E\): Let \(x \notin E\), we need to show that there is a function \(a \in \KI(E)\) with \(a(x) \neq 0\). This follows from the existence of bump function with value 1 in an open neighbourhood of \(x\) and \(0\) on \(E\). Here we use that \(E\) is closed, since we need to separate \(x\) and \(E\).
(\(\subset\)) Let \(f \in \KI(E)\), so \(f|_E = 0\). Then \(\phi \circ f\) is flat along \(E\), hence \(\phi \circ f \in I_\infty(E)\).
(\(\supset\)) Let \(f = g + r\) with \(g \in \sqrt[\lambda]{I}\) and \(r \in I_\infty(E)\). We have to show that \(f|_E = 0\), where \(E = Z(I)\). Pick \(\varphi \in \Lambda\) with \(\varphi \circ g \in I\). Now \(\varphi \circ f = \varphi \circ g + \varphi \circ r\). And \(\varphi \circ g|_E = 0\) since \(\varphi \circ g \in I\). At the same time \(\varphi \circ r|_E\) since it lies in \(r \in I_\infty(E)\). This shows \(\varphi \circ f|_E = 0\) and therefore \(f|_E=0\) as \(\varphi^{-1}(0) = 0\).
Corollary (Representation)
For a finitely generated ideal \(U\) with generators \(g_1, \dots, g_m \in \CINF(U)\), and a function \(f \in \mathcal{I}(E)\) there exist \(\varphi \in \Lambda\), smooth \(a_j\), and \(r \in I^\infty(E)\) such that: $$ \varphi \circ f = \sum_{j=1}^m a_j \, g_j + r, \quad r \in \KI_\infty(E). $$
Example (Necessity of both summands)
- Let \(I = (x^2 + y^2)\) then \(Z(I) = \{0\} \in \IR^2\). Now $x \in $
- Let \(I = (\varphi \circ \theta) \subset \AINF_1\) generated by the standard flat function, then \(Z(I)=0\). Note that the traditional radical \(rad(I) = \Set{f}{f^n \in I}\) does not contain the coordinate function \(x\). Also \(x\) does not lie in \(\KI_\infty(0) = \fF\). However \(x \in \sqrt[\Lambda]{I}\) since \(\phi \in \Lambda\).
- In the same setting, consider an oscillatory flat function with a wild zero set including \(0\). If \(\theta\) were in \(\sqrt[\Lambda]{I}\) we could find \(\varphi \in \Lambda, h \in \AINF_E\) so that \(\varphi \circ \theta = h \cdot \phi\). Equivalently: \(\varphi \circ \theta/ \phi\) extends as a smooth function through \(0\). For this to happen the product \(h \phi\) most be constant along the level sets of \(\theta\). So \(h\) must exactly vary to cancel out the restriction of \(\phi\) along each level set. There is no a priori smooth hh that can do this if the fibers are tangled enough. This is the natural place a counterexample should live.
Whitney Extensions
Definition
- For a point \(x \in U\), and a function \(f \in \CINF(U)\) we define:
- \(f_x \in \AINF_n\) to be the germ of \(f\) at \(x\) germs translated to the origin via \(\phi: y \mapsto y-x\): \(f_x = \phi^*f \in \AINF_n\).
- \(I_x \subset \AINF_n\) be the ideal generated by all \(f_x\) for \(f \in I\).
- \(T_k[I_x] \subset \IR[x^1,\dots,x^n]\) be the ideal generated by all degree-k taylor polynomials of functions in \(I_x\).
- \(T[I_x] \subset \FPOW\) be the ideal generated by all Taylor series of functions in \(I\).
Theorem (Borel). For every power series \(p \in \IR\dbrackets{x^1, \dots, x^n}\) there exists a \(C^\infty\) function \(a \in \AINF_n\) with \(T(a) = p\). In other words, the map \(T : \AINF_n \to \IR\dbrackets{x^1, \dots, x^n}\) is surjective, and hence \(\IR\dbrackets{x^1, \dots, x^n} \cong \AINF_n / \ker(T)\).
For a proof see Ieke Moerdijk, Gonzalo E. Reyes: Models for Smooth Infinitesimal Analysis, p13. or ncatlab. The basic idea is to compose \(a\) as an infinite linear combination of bump functions that are realizing higher taylor coefficients while ensuring convergence.
Tangent Vectors
In this section we construct tangent and co-tangent vectors spaces at the origin \(0 \in \IR^n\).
Tangent Space
- For an \(\AINF_n\)-module \(M\), the set of \(\IR\)-linear derivations \(Der_\IR(\AINF_n, M) \subset Hom_\IR(\AINF_n, M)\) is the set of all \(\IR\)-linear maps \(\delta: \AINF_n \ra M\) satisfying the product rule \(\delta[a \cdot b] = b \cdot_M \delta[a] + a \cdot_M \delta[b]\).
- Consider \(\IR\) as asn \(\AINF_n\) module via \(e[a] = a(0)\). The tangent space of \(\AINF_n\) at \(0\) is defined as set of derivation $$ T_0 := T_0^n := T_0 \AINF_n = Der_\IR(\AINF_n, \IR). $$
- The tangent space is a finite dimensional \(\IR\) vector space with basis \(dx_i := e \del_i a \mapsto \del_i[a](0)\): $$ T_0 = \IR< dx_1, \dots, dx_n>. $$
We will prove the last statement in the next paragraph.
Cotangent Space
The co-tangent space of \(\AINF_n\) at \(0\) is defined as \(\Omega_0 := \Omega_0^n := \Omega_0 \AINF_n := \fm/\fm^2\).
It comes with a canonical map \(d: \AINF_n \ra \Omega_0, a \mapsto a - \underline{ev_0[a]} = a - a(0)\), called the exterior differential.
- The exterior differential \(d: \AINF_n \ra \Omega_0\) is a derivation.
- The cotangent space is a finite dimensional \(\IR\) vector space with basis \(dx^i = d[x^i]\). $$ \Omega_0 = \IR< dx^1, \dots, dx^n> $$
- The cotangent space has the following universal property. For every \(\IR\) vectors space \(V\), the map $$ Hom_\IR(\Omega_0, V) \ra Der_\IR(\AINF_n, V), \alpha \mapsto \alpha \circ d $$ is an isomorphism.
- For \(\omega \in \Omega_0\) and \(\delta \in T_0\), we denote the natural pairing between tangent and co-tangent space as: $$ (\delta, \omega) = \delta[\omega] \in \IR $$
Proof
To show that the exterior differential is a derivation we calculate: \(d[a b] - b(0) d[a] - a(0) d[b] = \dots = - d[a] \cdot d[b] = 0 \in \fm/\fm^2\).
Let \(\delta\) be a derivation. Note that \(\delta[1] = \delta[1 1] = 2 \delta[1]\), hence \(\delta[1] = 0\). For \(a,b \in \fm\) we have \(d[a b] = \delta[a]e[b] + e[a] \delta[b] = 0\), hence \(\delta = 0\) on \(\fm^2\). This shows that \(\delta\) factors through \(d: \AINF_n \ra \fm/\fm^2\).
Conversely, every linear map \(\alpha: \fm/\fm^2 \ra \IR\) defines a derivation \(\alpha \circ d\). This shows the universal property of the co-tangent space.
By Hadamard's lemma we have \(\Omega_0 = \fm/\fm^2 = \IR<dx^1, \dots, dx^n>.\). Hence \(T_0 \isom Der_\IR(\AINF_n, \IR) \isom Hom_\IR(\Omega_0, \IR)\). Under this identification \(dx_i\) and \(dx^i\) are dual to each other: \(dx_i[dx^j] = e \del_i[x^j] = \delta_{i,j}\).
Naturality.
- A differentiable map \(\vphi: \IR^n \lra \IR^m\) with \(\vphi(0) = 0\) induces a linear map by pre-composition of derivations: $$ D_0\vphi = \vphi_* : T_0^n \lra T_0^m, \quad \delta \mapsto \delta \circ \vphi $$ and similarly on the cotangent space $$ D_0^*\vphi = \vphi^*: \Omega_0^m \lra \Omega^n_0, \quad b \mapsto b \circ \vphi. $$
- Naturality (Chain Rule). If \(\psi: \IR^m \lra \IR^l\) is a second differentiable map with \(\psi(0) = 0\), then $$ D_0(\psi \circ \vphi) = D_0\psi \circ D_0\vphi, \quad D_0^*(\psi \circ \vphi) = D_0^*\vphi \circ D_0^*\psi, $$ This statement is equivalent to the claim that \(D_0,D_0^*\) are natural transformations, which is obvious from the definition.
- Those maps are dual/adjoint to each other with regards to the isomorphism \(T_0 \isom Hom(\Omega_0, \IR)\). For \(\omega \in \Omega_0^m\) and \(\delta \in T_0^n\) we have: $$ (\vphi_*(\delta), \omega) = (D_0 \vphi(\delta), \omega) = (\delta, D_0^*\vphi(\omega)) = (\delta,\vphi^* \omega) $$ If \(\omega = db\) then all those expressions are equal to \(\delta(b \circ \vphi)\).
- Linear maps. If \(A: \IR^n \ra \IR^m\) is a linear map, then $$ dy^j(D_0 A)(dx_i) = y^j A x_i = A_{ij}. $$ Hence "\(D_0A = A\)" under the identification of \(T_0^n \isom \IR^n\) given by the canonical basis \(dx_i\).
- Coordinate representation. In coordinates \(x^i,x_i\) on \(\IR^n\) and \(y^j,y_j\) on \(\IR^m\) we have: $$ D_0 \vphi = D_0( (\sum_j y_j \circ y^j) \circ \vphi( \sum_i x_i \circ x^i ) ) = \sum_{j} D_0 y_j \circ D_0(y^j \circ \vphi) \circ D_0( \sum_i x_i \circ x^i ) $$ $$ = \sum_{i,j} D_0 y_j \circ D_0(y^j \circ \vphi \circ x_i) \circ D_0(x^i) $$ $$ = \sum_{i,j} \frac{\del \vphi^j}{\del x^i}(0) \cdot dy_j \circ dx^i $$ Where we used the canonical identifications \(dx_i = (D_0 x_i)(e \del_t)\), \(dx^i = D_0 x^i\). Also note that, while \(\vphi\) is not linear, the derivative \(D_0 \vphi\) is linear and we can "pull-out" summation in step 3.
Implicit Functions
Decay Lemma.
Let \(N \in \mathcal{E}_n^{\vsum n}\), a vector-valued germ vanishing to second order, i.e. \(N^i \in \mathfrak{m}_n^2\). Let \(a \in \mathcal{E}_k\) with \(a(0) = 0\) (i.e. \(a \in \mathfrak{m}_k\)).
Define the pullback \(N^*a = a \circ N \in \mathcal{E}_n\).
Then for every integer \(r \geq 0\), there exist constants \(C_r > 0\), \(\delta_0 > 0\) such that for all \(0 < \delta < \delta_0\), we have:
Proof
We proceed by induction on \(r\).
Case \(r = 0\):
Since \(a \in \mathfrak{m}_k\), Hadamard's lemma gives: \(a(y) = \sum_{i=1}^k y_i a_i(y)\), where \(a_i \in \mathcal{E}_k\). Then: \(a(N(x)) = \sum_{i=1}^k N_i(x) a_i(N(x))\). Since \(N_i(x) = \mathcal{O}(|x|^2)\), and \(a_i\) is smooth near 0, we get: \(|a(N(x))| \leq C |x|^2 \Rightarrow \| a \circ N \|_{C^0(B_\delta)} \leq C \delta^2\).
Case \(r = 1\):
By the chain rule: \(\partial_j (a \circ N)(x) = \sum_{i=1}^k \partial_i a(N(x)) \cdot \partial_j N_i(x)\). Since \(\partial_i a(N(x)) = \mathcal{O}(|x|^2)\) and \(\partial_j N_i(x) = \mathcal{O}(|x|)\), we get: \(|\partial_j (a \circ N)(x)| \leq C |x|^3 \Rightarrow \| \nabla(a \circ N) \|_{C^0(B_\delta)} \leq C \delta^3\). Thus: \(\| a \circ N \|_{C^1(B_\delta)} \leq C \delta^2\) (since the \(C^0\)-term dominates).
Inductive Step:
The higher derivatives involve terms of the form: \(\partial^\alpha(a \circ N)(x) = \sum \text{products of derivatives of } a \text{ and } N\). Each term gains at least quadratic vanishing from \(N\), and composition does not increase vanishing order. So: \(\| a \circ N \|_{C^r(B_\delta)} \leq C_r \delta^2 \cdot \| a \|_{C^r(B_{C \delta^2})}\).
Convergence Theorem for the von Neumann Series.
Let \(N \in \mathcal{E}_n^{\vsum n}\), a vector-valued germ vanishing to second order, i.e. \(N^i \in \mathfrak{m}_n^2\). Then the iterated compositions \(N^{[k]} := N \circ \cdots \circ N \in \mathcal{E}_n^n\) converge (componentwise) in the topology of \(\mathcal{E}_n^n\) to a germ:
Furthermore if \(F := \operatorname{id} - N \in \mathcal{E}_n^n\) then \(G\) is the inverse of \(F\), i.e.: \(F \circ G = \operatorname{id}, G \circ F = \operatorname{id}\).
Proof
We use the decay lemma componentwise. Let \(x_i \in \mathfrak{m}_n\) be the coordinate functions. Define: $$ g_i := \sum_{k=0}^\infty x_i \circ N^{[k]} $$ Each term \(x_i \circ N^{[k]} \in \mathfrak{m}_n^{2^k}\), so: $$ | x_i \circ N^{[k]} |{C^r(B\delta)} \leq C_r \delta^{2^k} $$ which implies convergence in \(\mathcal{E}_n\) topology for each \(g_i\). Let: $$ G := (g_1, \dots, g_n) \in \mathcal{E}_n^n $$
To verify \(F \circ G = \operatorname{id}\), note: $$ F \circ G = (\operatorname{id} - N) \circ \left( \sum_{k=0}^\infty N^{[k]} \right) = \sum_{k=0}^\infty (N^{[k]} - N^{[k+1]}) = \operatorname{id} $$ Similar computation shows \(G \circ F = \operatorname{id}\).
Differential Equations
Definition
Consider a function in a single real variable \(a \in \AINF_1\). Define the integration operator \(I: \AINF_1 \ra \AINF_1\), by
Here \(t\) is a variable varying in the domain of definition \(0 \in U \subset \IR\) of \(a\).
Lemma (Integration estimate on \(\mathfrak{m}^k\) in \(C^k\)-norm)
Let \(k \in \IN\), and let \(a \in C^k((-\eps,\eps))\) be a function satisfying \(a \in \mathfrak{m}^k\). Then there exists a \(\delta_0\) so that for all \(\delta < \delta_0\) we have:
for some constant \(C_k > 0\) depending only on \(k\), but not on \(a\) or \(\delta\).
Proof
Since \(a \in \mathfrak{m}^k\), by Hadamard's lemma we can write \(a(t) = t^k \cdot a'(t)\) for some function \(a' \in C^0([0, \delta])\). Then \(I[a](t) = \int_0^t s^k \cdot a'(s)\, ds\).
We estimate the \(C^k\)-norm of \(I[a]\) on the interval \([0, \delta]\). Let's consider the derivatives:
- For \(j = 0\): We have \(|I[a](t)| \leq \|a'\|_\infty \cdot \int_0^t s^k\, ds = \|a'\|_\infty \cdot \frac{t^{k+1}}{k+1}\), hence \(\|I[a]\|_{0,\delta} \leq \frac{\delta^{k+1}}{k+1} \cdot \|a'\|_{C^0}\).
-
For \(1 \leq j \leq k\), we differentiate under the integral sign: \(\frac{d^j}{dt^j} I[a](t) = a^{(j-1)}(t)\).
To estimate \(a^{(j-1)}(t)\), we apply the Leibniz rule to the product \(a(t) = t^k \cdot a'(t)\):
\(a^{(j-1)}(t) = \sum_{i=0}^{j-1} \binom{j-1}{i} \cdot \frac{d^i}{dt^i}(t^k) \cdot \frac{d^{j-1 - i}}{dt^{j-1 - i}} a'(t)\).
Each term \(\frac{d^i}{dt^i}(t^k)\) is a polynomial in \(t\), bounded on \([0, \delta]\) by a constant times \(\delta^{k - i}\). Thus: \(\|a^{(j-1)}\|_{C^0} \leq C_{k,j} \cdot \sum_{i=0}^{j-1} \delta^{k - i} \cdot \|a'\|_{C^{j-1 - i}}\).
In particular, for \(j \leq k\), we obtain: \(\left\| \frac{d^j}{dt^j} I[a] \right\|_{C^0} \leq C'_{k,j} \cdot \delta \cdot \|a'\|_{C^{k-1}}\).
Putting all together, we have: \(\|I[a]\|_{k,\delta} = \max_{0 \leq j \leq k} \left\| \frac{d^j}{dt^j} I[a] \right\|_{C^0} \leq C_k \cdot \delta \cdot \|a'\|_{C^{k-1}}\).
Finally, since \(a(t) = t^k a'(t)\), applying the Leibniz rule in reverse gives: \(\|a\|_{k,\delta} \leq D_k \cdot \|a'\|_{C^k}\), so in particular: \(\|a'\|_{C^{k-1}} \leq D_k' \cdot \|a\|_{k,\delta}\).
Hence, combining constants: \(\|I[a]\|_{k,\delta} \leq \delta \cdot C_k \cdot \|a\|_{k,\delta}\), as claimed.
BACKUP
Implicit Functions
Germs of Functions. The category \(G\) of germs of \(C^\infty\)-spaces has objects \(\IR^n_{,0}\) for \(n \in \IN_0\) and morphisms germs of \(C^\infty\)-functions:
where \(C^\infty_0(U, \IR^m)\) is the set of \(C^\infty\)-functions \(\IR^n \ra \IR^m\) mapping \(0\) to \(0\). Composition of functions descends to an associative composition operations on \(G\).
Properties
- Every function \(\vphi: \IR^n \ra \IR^m\) with \(\vphi(0) = 0\) defines a germ \(\vphi \in Mor(\IR^n_{,0},\IR^m_{,0})\).
- Elements in \(Mor(\IR^n_{,0},\IR^m_{,0})\) are represented by functions \(\vphi: U \ra \IR^n\) defined in an open neighborhood \(U\) of \(0 \in \IR^n\).
- Germs \(\vphi \in Mor(\IR^n_{,0},\IR^m_{,0})\) induce algebra morphisms \(\vphi^*: \AINF_m \ra \AINF_n\).
Theorem (Inverese Functions).
If \(\vphi: \IR^n_{,0} \ra \IR^n_{,0}\) is a function germ, then \(\vphi\) is invertible as a germ if \(D_0 \vphi\) is invertible as a linear endomorphism of \(T_0\). In this case \(D_0 \vphi^{-1} = (D_0\vphi)^{-1}\).
Theorem (Implicit Function).
Let \(a \in \AINF_n\) be a function with \(a(0) = 0\) and \(dx_n \cdot a \neq 0\), then:
- There exists a function \(\vphi: \IR^{n-1}_{,0} \ra \IR^{n}_{,0}\), with \(a(x_1, \dots, x_n) = 0\) if and only if \(x_n = \vphi(x_1,\dots,x_{n-1}).\)
- The projetion \(\pi: \IR^{n} \ra \IR^{n-1}, (x_1,\dots,x_n) \mapsto (x_2, \dots, x_n)\) induces an isomorphism \(\pi^*: \AINF^{n-1} \ra \AINF_n /(a)\).
Invariant Theory
Group Action. Let \(\mathrm{Diff}_0^n\) be the group of local diffeomorphisms of \(\IR^n\). Elements \(\varphi \in \mathrm{Diff}_0^n\) are represented by diffeomorophisms \(\varphi: U \ra V\), with \(0 \in U, 0 \in V\). Two representatives \(\varphi: U \ra V, \varphi': U' \ra V'\) are equivalent if they agree on \(U \cap U'\).
Evenry element \(\varphi \in \mathrm{Diff}_0^n\) gives rise to an liear isomorphism \(\varphi^*\) of \(\AINF_n\) via the pullback operation. In this way we optain a group action \(\mathrm{Diff}_0^n \ra GL(\AINF_n), \varphi \mapsto \varphi^*\).
Question: For any any natural "structure" \(F\) on \(\AINF_n\), classify elements of \(F\) up to diffeomorphism.
Examples:
- \(F = id\): Classify local functions up to diffeomorphism. In the case \(n=1\) we claim that \(\AINF^1 / \mathrm{Diff}_0^1 = c(\IR) \union \\{ x^1 \\}\)
- \(F = T\) (Tangent space). Classify tangent vectors up to diffeopmorphis. We have \(T(\AINF_n) / \mathrm{Diff}_0^n = \\{ 0, \del_1 \\}\), reflecting the fact that two non-zero tangent vector can be transformed into each other via a (linear) diffeomorphism.
Topology and Convergence
We have defined \(\AINF_n\) as inverse limit over \(\CINF(U)\) where \(U \ni 0\). The spaces \(\CINF(U,\IR)\) are defined as intersections of all \(\mathcal{C}^k(U)\) where \(k \geq 0\). The spaces \(\mathcal{C}^k(U)\) come with a family of semi-norms \(\|f\|_{K,k} = \sup_{|\alpha| \leq k, x \in K} |\del^{\alpha} f|\) where \(K \subset U\) is compact. These norms induce a topology on \(\mathcal{C}^k(U)\) making it a complete metrizable locally convex topological vector space (a Fréchet space).
The space \(\CINF(U,\IR)\) is a projective limit of all \(\mathcal{C}^k(U)\) along the restrictions and inherits a natural topology making the space initial (in the topological category) for the projective system. This topology is the Frechet topology induced by the set of seminorms \(\|f\|_{K,k}\) where \(K,k\) are allowed to range freely.
The topology on \(\AINF_n\) is the final topology for the inductive system \(\{ \mathcal{C}^\infty(U) \}_{U \ni 0}\), where the transition maps are restrictions. A subset \( \mathcal{O} \subset \mathcal{A}^\infty_n \) is open if and only if there exists an open neighborhood \( U \ni 0 \) in \( \mathbb{R}^n \), and an open subset \(\mathcal{O}_U \subset \mathcal{C}^\infty(U)\), such that: \(\mathcal{O} = \{ [f] \in \mathcal{A}^\infty_n \mid f \in \mathcal{O}_U \}.\)
Proposition
The topology on \(\AINF_n\) is Haussdorff and complete.
Proof
Hausdorff
Let \( a, b \in \AINF_n \) be two distinct germs. Then \( a \ne b \) implies that there exists a representative \( f \in a \), \( g \in b \), and an open neighborhood \( U \ni 0 \) such that \( f, g \in \CINF(U) \) but \( f \ne g \) as functions on \( U \). In particular, \( f - g \ne 0 \) on some compact set \( K \subset U \).
Let \( \mathcal{O}_U(f) := \{ h \in \CINF(U) \mid \|h - f\|_{K,k} < \varepsilon/2 \} \), and define \(O_f := \{ [h] \in \AINF_n \mid h \in \mathcal{O}_U(f) \}\) and \(O_g := \{ [h] \in \AINF_n \mid h \in \mathcal{O}_U(g) \}\). These are open neighborhoods of \( a \) and \( b \) in \( \AINF_n \), respectively, and they are disjoint: any representative of a germ in \( O_f \) is within \( \varepsilon/2 \) of \( f \), and any representative in \( O_g \) is within \( \varepsilon/2 \) of \( g \), so their difference is strictly greater than \( \varepsilon - \varepsilon = 0 \). Hence \( a \ne b \Rightarrow \exists \) disjoint open neighborhoods, and \( \AINF_n \) is Hausdorff.
Completeness
Let \( (a_n)_{n \in \mathbb{N}} \subset \AINF_n \) be a Cauchy sequence. By definition of the inductive limit topology, this means: There exists an open neighborhood \( U \ni 0 \) such that all \( a_n \) admit representatives \( f_n \in \CINF(U) \), and \( (f_n) \) is a Cauchy sequence in the Fréchet space \( \CINF(U) \). Since \( \CINF(U) \) is complete, the sequence \( f_n \to f \in \CINF(U) \). Then \( [f_n] \to [f] \in \AINF_n \), so the sequence of germs \( a_n \) converges. Therefore, every Cauchy sequence in \( \AINF_n \) converges, and the space is complete.
Let \(C^k(\IR^n,0)\) be the inductive limit of \(C^k(U)\) over all open subsets \(\IR^n \supset U \ni 0\) along the restrictions. Let \(\mathcal{B}_n\) be the projective limit of \(C^k(\IR^n,0)\) along the inclusions \(C^{k+1}(\IR^n,0) \subset C^k(\IR^n,0)\) with it's induced topology.
Proposition
Then the natural maps \(\AINF_n \to C^k(\IR^n,0)\) induce a homeomorphism:
Proof
First validate that the map is a bijection of sets.
Every element of \(\AINF_n\) is a smooth germ \([f]\) at the origin, represented by some \(f \in \CINF(U)\) for \(U \ni 0\). For each \(k\), the restriction of \(f\) defines a class \([f]_k \in C^k(\IR^n,0)\) via the inclusion \(\CINF(U) \to \mathcal{C}^k(U) \to C^k(\IR^n,0).\) These are compatible under the projections \(C^{k+1}(\IR^n,0) \to C^k(\IR^n,0)\), so the germ \([f] \in \AINF_n\) defines an element in the projective limit \(([f]_k)_{k \in \mathbb{N}} \in \mathcal{B}_n.\)
Conversely, let \(([f_k]) \in \mathcal{B}_n\), where each \(f_k \in \mathcal{C}^k(U_k)\) for some neighborhood \(U_k \ni 0\), and the restrictions are compatible under \(k\). By shrinking to a common \(U \ni 0\), we may view all \(f_k\) as functions on \(U\), and the compatibility ensures that \(f_{k+1}\) agrees with \(f_k\) as a \(C^k\) function. Hence \(f = f_k\) lies in \(\CINF(U)\) and defines an element in \(\AINF_n\).
Now consider the topologies.
- A neighborhood basis in \(\AINF_n\) is given by sets of the form: \(\mathcal{O}_U(0) := \{ [f] \in \AINF_n \mid \|f\|_{K,k} < \varepsilon \},\) for some \(U \ni 0\), compact \(K \subset U\), and \(k \in \mathbb{N}\).
- A basic neighborhood in \(\mathcal{B}_n = \lim_{\la} C^k(\IR^n,0)\) is a subset of the form \(\mathcal{O}_{k,K,\varepsilon} := \{ ([f_j]) \in \mathcal{B}_n \mid \|f_k\|_{K,k} < \varepsilon \}\) for some \(k\), compact \(K \subset \IR^n\), and \(\varepsilon > 0\).
Under the identification above, the pullback of \(\mathcal{O}_{k,K,\varepsilon}\) to \(\AINF_n\) is exactly the set of germs \([f]\) with a representative \(f\) satisfying \(\|f\|_{K,k} < \varepsilon\), i.e., of the form \(\mathcal{O}_U(0)\) with \(K \subset U\).
Thus, the topologies coincide, and the map is a homeomorphism.