Equivalence of definitions for ring of germs $C_p^{infty}(mathbb R^n)$












1












$begingroup$


I want to show the characterizations of $C_p^{infty}(mathbb R^n)$ are equivalent:



$$A= {[f] | text{smooth} f: mathbb R^n to mathbb R }$$



$$B= {[f] | text{smooth} f: U_p to mathbb R }$$



$$C= {[f] | text{smooth} f: V_{f,p} to mathbb R }$$



where




  • all the germs in $B$ can be represented by functions whose domains are a fixed open subset $U_p$ of $mathbb R^n$ that contains $p$


  • while the domaina of functions that represent germs in $C$ are any open subsets $V_{f,p}$ of $mathbb R^n$ that contain $p$.



So far I have done:



$(A subseteq B):$




  • Let $[f] in A$ for a smooth $f: mathbb R^n to mathbb R$. We must show $[f] in B$, which is done by finding some smooth $g: U_p to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq mathbb R^n cap U_p = U_p$ of $mathbb R^n$ that contains $p$.


  • We can choose $g = f|_{U_p}$ and $W_p = U_p$.



$(B subseteq C):$




  • Let $[f] in B$ for a smooth $f: mathbb U_p to mathbb R$. We must show $[f] in C$, which is done by finding some smooth $g: V_{p,g} to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,g} cap U_p$ of $mathbb R^n$ that contains $p$.


  • We can choose $V_{p,g} = U_p$ and $g = f$.



$(C subseteq A):$




  • According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $mathbb R^n$ for $n=1$, doesn't always exist. However, it exists to say $C subseteq A$:


  • Let $[f] in C$ for a smooth $f: mathbb V_{p,f} to mathbb R$. We must show $[f] in A$, which is done by finding some smooth $g: mathbb R^n to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,f} cap mathbb R^n = V_{p,f}$ of $mathbb R^n$ that contains $p$.


  • We can choose $g = rho f 1_{U_p}$, where $rho: mathbb R^n to mathbb R$ is a smooth bump function supported in $U_p$ and $rho(D_p)={1}$ for some open subset $D_p$ of $mathbb R^n$ containing $p$, and we can choose $W_p=D_p$.



Questions




  1. Is anything wrong?


  2. Can $C subseteq A$ be proven without bump functions?







differential-geometry smooth-manifolds smooth-functions germs







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I want to show the characterizations of $C_p^{infty}(mathbb R^n)$ are equivalent:



    $$A= {[f] | text{smooth} f: mathbb R^n to mathbb R }$$



    $$B= {[f] | text{smooth} f: U_p to mathbb R }$$



    $$C= {[f] | text{smooth} f: V_{f,p} to mathbb R }$$



    where




    • all the germs in $B$ can be represented by functions whose domains are a fixed open subset $U_p$ of $mathbb R^n$ that contains $p$


    • while the domaina of functions that represent germs in $C$ are any open subsets $V_{f,p}$ of $mathbb R^n$ that contain $p$.



    So far I have done:



    $(A subseteq B):$




    • Let $[f] in A$ for a smooth $f: mathbb R^n to mathbb R$. We must show $[f] in B$, which is done by finding some smooth $g: U_p to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq mathbb R^n cap U_p = U_p$ of $mathbb R^n$ that contains $p$.


    • We can choose $g = f|_{U_p}$ and $W_p = U_p$.



    $(B subseteq C):$




    • Let $[f] in B$ for a smooth $f: mathbb U_p to mathbb R$. We must show $[f] in C$, which is done by finding some smooth $g: V_{p,g} to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,g} cap U_p$ of $mathbb R^n$ that contains $p$.


    • We can choose $V_{p,g} = U_p$ and $g = f$.



    $(C subseteq A):$




    • According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $mathbb R^n$ for $n=1$, doesn't always exist. However, it exists to say $C subseteq A$:


    • Let $[f] in C$ for a smooth $f: mathbb V_{p,f} to mathbb R$. We must show $[f] in A$, which is done by finding some smooth $g: mathbb R^n to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,f} cap mathbb R^n = V_{p,f}$ of $mathbb R^n$ that contains $p$.


    • We can choose $g = rho f 1_{U_p}$, where $rho: mathbb R^n to mathbb R$ is a smooth bump function supported in $U_p$ and $rho(D_p)={1}$ for some open subset $D_p$ of $mathbb R^n$ containing $p$, and we can choose $W_p=D_p$.



    Questions




    1. Is anything wrong?


    2. Can $C subseteq A$ be proven without bump functions?







    differential-geometry smooth-manifolds smooth-functions germs







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I want to show the characterizations of $C_p^{infty}(mathbb R^n)$ are equivalent:



      $$A= {[f] | text{smooth} f: mathbb R^n to mathbb R }$$



      $$B= {[f] | text{smooth} f: U_p to mathbb R }$$



      $$C= {[f] | text{smooth} f: V_{f,p} to mathbb R }$$



      where




      • all the germs in $B$ can be represented by functions whose domains are a fixed open subset $U_p$ of $mathbb R^n$ that contains $p$


      • while the domaina of functions that represent germs in $C$ are any open subsets $V_{f,p}$ of $mathbb R^n$ that contain $p$.



      So far I have done:



      $(A subseteq B):$




      • Let $[f] in A$ for a smooth $f: mathbb R^n to mathbb R$. We must show $[f] in B$, which is done by finding some smooth $g: U_p to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq mathbb R^n cap U_p = U_p$ of $mathbb R^n$ that contains $p$.


      • We can choose $g = f|_{U_p}$ and $W_p = U_p$.



      $(B subseteq C):$




      • Let $[f] in B$ for a smooth $f: mathbb U_p to mathbb R$. We must show $[f] in C$, which is done by finding some smooth $g: V_{p,g} to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,g} cap U_p$ of $mathbb R^n$ that contains $p$.


      • We can choose $V_{p,g} = U_p$ and $g = f$.



      $(C subseteq A):$




      • According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $mathbb R^n$ for $n=1$, doesn't always exist. However, it exists to say $C subseteq A$:


      • Let $[f] in C$ for a smooth $f: mathbb V_{p,f} to mathbb R$. We must show $[f] in A$, which is done by finding some smooth $g: mathbb R^n to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,f} cap mathbb R^n = V_{p,f}$ of $mathbb R^n$ that contains $p$.


      • We can choose $g = rho f 1_{U_p}$, where $rho: mathbb R^n to mathbb R$ is a smooth bump function supported in $U_p$ and $rho(D_p)={1}$ for some open subset $D_p$ of $mathbb R^n$ containing $p$, and we can choose $W_p=D_p$.



      Questions




      1. Is anything wrong?


      2. Can $C subseteq A$ be proven without bump functions?







      differential-geometry smooth-manifolds smooth-functions germs







      share|cite|improve this question











      $endgroup$




      I want to show the characterizations of $C_p^{infty}(mathbb R^n)$ are equivalent:



      $$A= {[f] | text{smooth} f: mathbb R^n to mathbb R }$$



      $$B= {[f] | text{smooth} f: U_p to mathbb R }$$



      $$C= {[f] | text{smooth} f: V_{f,p} to mathbb R }$$



      where




      • all the germs in $B$ can be represented by functions whose domains are a fixed open subset $U_p$ of $mathbb R^n$ that contains $p$


      • while the domaina of functions that represent germs in $C$ are any open subsets $V_{f,p}$ of $mathbb R^n$ that contain $p$.



      So far I have done:



      $(A subseteq B):$




      • Let $[f] in A$ for a smooth $f: mathbb R^n to mathbb R$. We must show $[f] in B$, which is done by finding some smooth $g: U_p to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq mathbb R^n cap U_p = U_p$ of $mathbb R^n$ that contains $p$.


      • We can choose $g = f|_{U_p}$ and $W_p = U_p$.



      $(B subseteq C):$




      • Let $[f] in B$ for a smooth $f: mathbb U_p to mathbb R$. We must show $[f] in C$, which is done by finding some smooth $g: V_{p,g} to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,g} cap U_p$ of $mathbb R^n$ that contains $p$.


      • We can choose $V_{p,g} = U_p$ and $g = f$.



      $(C subseteq A):$




      • According to a proposition in Section 13 (11 sections later), an arbitrary smooth extension to the whole manifold, even in the case of $mathbb R^n$ for $n=1$, doesn't always exist. However, it exists to say $C subseteq A$:


      • Let $[f] in C$ for a smooth $f: mathbb V_{p,f} to mathbb R$. We must show $[f] in A$, which is done by finding some smooth $g: mathbb R^n to mathbb R$ such that $[f]=[g]$, which means $f|_{W_p} = g|_{W_p}$ for an a open subset $W_p subseteq V_{p,f} cap mathbb R^n = V_{p,f}$ of $mathbb R^n$ that contains $p$.


      • We can choose $g = rho f 1_{U_p}$, where $rho: mathbb R^n to mathbb R$ is a smooth bump function supported in $U_p$ and $rho(D_p)={1}$ for some open subset $D_p$ of $mathbb R^n$ containing $p$, and we can choose $W_p=D_p$.



      Questions




      1. Is anything wrong?


      2. Can $C subseteq A$ be proven without bump functions?







      differential-geometry smooth-manifolds smooth-functions germs




      differential-geometry smooth-manifolds smooth-functions germs






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 19 at 2:38







      Selene Auckland

















      asked Jan 26 at 8:31









      Selene AucklandSelene Auckland

      7911




      7911






















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