Mixing
Equivalence of definitions for ring of germs $C_p^{infty}(mathbb R^n)$
$begingroup$
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
Is anything wrong?
Can $C subseteq A$ be proven without bump functions?
differential-geometry smooth-manifolds smooth-functions germs
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
$endgroup$
add a comment |
$begingroup$
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
Is anything wrong?
Can $C subseteq A$ be proven without bump functions?
differential-geometry smooth-manifolds smooth-functions germs
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
$endgroup$
add a comment |
$begingroup$
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
Is anything wrong?
Can $C subseteq A$ be proven without bump functions?
differential-geometry smooth-manifolds smooth-functions germs
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
$endgroup$
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
Is anything wrong?
Can $C subseteq A$ be proven without bump functions?
differential-geometry smooth-manifolds smooth-functions germs
differential-geometry smooth-manifolds smooth-functions germs
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
edited Feb 19 at 2:38
Selene Auckland
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
asked Jan 26 at 8:31
Selene AucklandSelene Auckland
7911
7911
add a comment |
add a comment |
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