Mixing
Infinite dimensional tangent spaces
$begingroup$
Let $p in mathbb{R}^n$ . It is well-known that there are at least two equivalent definitions of the tangent space at $p$:
$T_p mathbb{R}^n$ is the set of equivalence classes of $C^1$-curves $gamma : (-1,1) to mathbb{R}^n$ with $gamma(0)=p$ and where two curves are equivalent, iff they share the same derivative at $0$.
(The Zariski way) $T_p mathbb{R}^n$ is the set of all derivations $D : C^infty(mathbb{R}^n,mathbb{R})_p to mathbb{R}$ on the stalk at $p$ of the sheaf of smooth functions.
The trick to prove the equivalence of both definitions is the fact, that for every smooth function $f: U to mathbb{R}$, $U ni p$ open, there are $varepsilon <0$, smooth functions $f_1, dots , f_n$ with $f_i(p)= frac{partial f}{partial x^i}(p)$ and
$$ f(x^1, dots , x^n) = sum_{i=1}^n x^i cdot f_i(x^1, dots , x^n) qquad (x^1, dots, x^n) in mathbb{B}_varepsilon(p)$$
The $f_i$ are defined via
$$ f_i : x mapsto int_0^1 frac{partial f}{partial x^i}(p + t(x-p) dt $$
Note that, if $f in C^n(mathbb{R}^n,mathbb{R})_p$, the functions $f_i$ are only $C^{n-1}$, so this trick would fail if being used for $C^n$-manifolds.
Now let $E$ be an arbitrary real Banach space and $p in E$. As before, we have at least two options to define the tangent space $T_pE$. Every equivalence class of curves, sharing the same derivative, yields a derivation. But I see no proof for the converse. If $f : U to mathbb{R}$ is a smooth function on an open neighborhood of $p$, there exists an $varepsilon >0$ and a smooth function $A : mathbb{B}_varepsilon(p) to E'$, $E'$ being the continuous dual space of $E$, such that
$$ f(x) = langle A(x), x rangle qquad x in mathbb{B}_varepsilon(p)$$
and $A(p)=df(p)$ where $df : E to E'$ is the Frechet-differential of $f$. $A$ is defined via a Bochner integral
$$ A :x mapsto int_0^1 df(p+t(x-p)) dt $$
At this point, the product rule for derivations cannot help us.
Note that the curve-definition leads to a tangent space isomorphic to $E$ while I guess the derivation space to be "larger".
My question: How to define $T_pE$? And how to do it, if $E$ is a complex Banach space?
banach-spaces tangent-spaces
asked Jan 15 at 10:53
LucinaLucina
605
$endgroup$
add a comment |
$begingroup$
Let $p in mathbb{R}^n$ . It is well-known that there are at least two equivalent definitions of the tangent space at $p$:
$T_p mathbb{R}^n$ is the set of equivalence classes of $C^1$-curves $gamma : (-1,1) to mathbb{R}^n$ with $gamma(0)=p$ and where two curves are equivalent, iff they share the same derivative at $0$.
(The Zariski way) $T_p mathbb{R}^n$ is the set of all derivations $D : C^infty(mathbb{R}^n,mathbb{R})_p to mathbb{R}$ on the stalk at $p$ of the sheaf of smooth functions.
The trick to prove the equivalence of both definitions is the fact, that for every smooth function $f: U to mathbb{R}$, $U ni p$ open, there are $varepsilon <0$, smooth functions $f_1, dots , f_n$ with $f_i(p)= frac{partial f}{partial x^i}(p)$ and
$$ f(x^1, dots , x^n) = sum_{i=1}^n x^i cdot f_i(x^1, dots , x^n) qquad (x^1, dots, x^n) in mathbb{B}_varepsilon(p)$$
The $f_i$ are defined via
$$ f_i : x mapsto int_0^1 frac{partial f}{partial x^i}(p + t(x-p) dt $$
Note that, if $f in C^n(mathbb{R}^n,mathbb{R})_p$, the functions $f_i$ are only $C^{n-1}$, so this trick would fail if being used for $C^n$-manifolds.
Now let $E$ be an arbitrary real Banach space and $p in E$. As before, we have at least two options to define the tangent space $T_pE$. Every equivalence class of curves, sharing the same derivative, yields a derivation. But I see no proof for the converse. If $f : U to mathbb{R}$ is a smooth function on an open neighborhood of $p$, there exists an $varepsilon >0$ and a smooth function $A : mathbb{B}_varepsilon(p) to E'$, $E'$ being the continuous dual space of $E$, such that
$$ f(x) = langle A(x), x rangle qquad x in mathbb{B}_varepsilon(p)$$
and $A(p)=df(p)$ where $df : E to E'$ is the Frechet-differential of $f$. $A$ is defined via a Bochner integral
$$ A :x mapsto int_0^1 df(p+t(x-p)) dt $$
At this point, the product rule for derivations cannot help us.
Note that the curve-definition leads to a tangent space isomorphic to $E$ while I guess the derivation space to be "larger".
My question: How to define $T_pE$? And how to do it, if $E$ is a complex Banach space?
banach-spaces tangent-spaces
asked Jan 15 at 10:53
LucinaLucina
605
$endgroup$
$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41
add a comment |
$begingroup$
Let $p in mathbb{R}^n$ . It is well-known that there are at least two equivalent definitions of the tangent space at $p$:
$T_p mathbb{R}^n$ is the set of equivalence classes of $C^1$-curves $gamma : (-1,1) to mathbb{R}^n$ with $gamma(0)=p$ and where two curves are equivalent, iff they share the same derivative at $0$.
(The Zariski way) $T_p mathbb{R}^n$ is the set of all derivations $D : C^infty(mathbb{R}^n,mathbb{R})_p to mathbb{R}$ on the stalk at $p$ of the sheaf of smooth functions.
The trick to prove the equivalence of both definitions is the fact, that for every smooth function $f: U to mathbb{R}$, $U ni p$ open, there are $varepsilon <0$, smooth functions $f_1, dots , f_n$ with $f_i(p)= frac{partial f}{partial x^i}(p)$ and
$$ f(x^1, dots , x^n) = sum_{i=1}^n x^i cdot f_i(x^1, dots , x^n) qquad (x^1, dots, x^n) in mathbb{B}_varepsilon(p)$$
The $f_i$ are defined via
$$ f_i : x mapsto int_0^1 frac{partial f}{partial x^i}(p + t(x-p) dt $$
Note that, if $f in C^n(mathbb{R}^n,mathbb{R})_p$, the functions $f_i$ are only $C^{n-1}$, so this trick would fail if being used for $C^n$-manifolds.
Now let $E$ be an arbitrary real Banach space and $p in E$. As before, we have at least two options to define the tangent space $T_pE$. Every equivalence class of curves, sharing the same derivative, yields a derivation. But I see no proof for the converse. If $f : U to mathbb{R}$ is a smooth function on an open neighborhood of $p$, there exists an $varepsilon >0$ and a smooth function $A : mathbb{B}_varepsilon(p) to E'$, $E'$ being the continuous dual space of $E$, such that
$$ f(x) = langle A(x), x rangle qquad x in mathbb{B}_varepsilon(p)$$
and $A(p)=df(p)$ where $df : E to E'$ is the Frechet-differential of $f$. $A$ is defined via a Bochner integral
$$ A :x mapsto int_0^1 df(p+t(x-p)) dt $$
At this point, the product rule for derivations cannot help us.
Note that the curve-definition leads to a tangent space isomorphic to $E$ while I guess the derivation space to be "larger".
My question: How to define $T_pE$? And how to do it, if $E$ is a complex Banach space?
banach-spaces tangent-spaces
asked Jan 15 at 10:53
LucinaLucina
605
$endgroup$
Let $p in mathbb{R}^n$ . It is well-known that there are at least two equivalent definitions of the tangent space at $p$:
$T_p mathbb{R}^n$ is the set of equivalence classes of $C^1$-curves $gamma : (-1,1) to mathbb{R}^n$ with $gamma(0)=p$ and where two curves are equivalent, iff they share the same derivative at $0$.
(The Zariski way) $T_p mathbb{R}^n$ is the set of all derivations $D : C^infty(mathbb{R}^n,mathbb{R})_p to mathbb{R}$ on the stalk at $p$ of the sheaf of smooth functions.
The trick to prove the equivalence of both definitions is the fact, that for every smooth function $f: U to mathbb{R}$, $U ni p$ open, there are $varepsilon <0$, smooth functions $f_1, dots , f_n$ with $f_i(p)= frac{partial f}{partial x^i}(p)$ and
$$ f(x^1, dots , x^n) = sum_{i=1}^n x^i cdot f_i(x^1, dots , x^n) qquad (x^1, dots, x^n) in mathbb{B}_varepsilon(p)$$
The $f_i$ are defined via
$$ f_i : x mapsto int_0^1 frac{partial f}{partial x^i}(p + t(x-p) dt $$
Note that, if $f in C^n(mathbb{R}^n,mathbb{R})_p$, the functions $f_i$ are only $C^{n-1}$, so this trick would fail if being used for $C^n$-manifolds.
Now let $E$ be an arbitrary real Banach space and $p in E$. As before, we have at least two options to define the tangent space $T_pE$. Every equivalence class of curves, sharing the same derivative, yields a derivation. But I see no proof for the converse. If $f : U to mathbb{R}$ is a smooth function on an open neighborhood of $p$, there exists an $varepsilon >0$ and a smooth function $A : mathbb{B}_varepsilon(p) to E'$, $E'$ being the continuous dual space of $E$, such that
$$ f(x) = langle A(x), x rangle qquad x in mathbb{B}_varepsilon(p)$$
and $A(p)=df(p)$ where $df : E to E'$ is the Frechet-differential of $f$. $A$ is defined via a Bochner integral
$$ A :x mapsto int_0^1 df(p+t(x-p)) dt $$
At this point, the product rule for derivations cannot help us.
Note that the curve-definition leads to a tangent space isomorphic to $E$ while I guess the derivation space to be "larger".
My question: How to define $T_pE$? And how to do it, if $E$ is a complex Banach space?
banach-spaces tangent-spaces
banach-spaces tangent-spaces
asked Jan 15 at 10:53
LucinaLucina
605
asked Jan 15 at 10:53
LucinaLucina
605
asked Jan 15 at 10:53
LucinaLucina
605
asked Jan 15 at 10:53
LucinaLucina
605
asked Jan 15 at 10:53
LucinaLucina
605
605
$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41
add a comment |
$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41
$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41
$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41
add a comment |
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$begingroup$
Relevant: mathoverflow.net/questions/65926/….
$endgroup$
– Martín-Blas Pérez Pinilla
Jan 18 at 15:41