This question is apropos of a comment I received in the question: Lang Fundamentals of Differential Geomety definition of covariant derivative of a tensor field. The comment referred to a proposition in Abraham, Marsden, Ratiu, Manifolds, Tensor Analysis, and Applications (hereafter AMR). In the second edition, it is Proposition 5.2.20 and in the third edition it is Proposition 6.2.20. A complete understanding of the AMR proposition would be useful in addressing the referred to question.
Proposition. Let $M$ be a finite-dimensional manifold or be modeled on a Banach space with norm $C^\infty$ away from the origin. Then $\mathcal{T}^r_s(M)$ is isomorphic to $\mathfrak{T}^r_s(M)$ regarded as $\mathcal{F}(M)$-modules and as real vector spaces. In particular $\mathfrak{X}^*(M)$ is isomorphic to $\mathcal{X}^*(M)$.
[Before citing the proof, here are the definitions given just before the proposition:
$$L_{\mathcal{F}(M)}(\mathfrak{X}(M),\mathcal{F}(M))=\mathcal{X}^*(M)$$ the $\mathcal{F}(M)$-linear mappings on $\mathfrak{X}(M)$, and similarly \begin{equation*} \mathfrak{T}^r_s(M)=L^{r+s}_{\mathcal{F}(M)}(\mathcal{X}^*(M),\dots,\mathfrak{X}(M); \mathcal{F}(M)) \end{equation*} the $\mathcal{F}(M)$-multilinear mappings.
I'm pretty certain there is a typo (theirs, not mine) in that last display. It would make far more sense if the definition was \begin{equation*} \mathfrak{T}^r_s(M)=L^{r+s}_{\mathcal{F}(M)}(\mathfrak{X}^*(M),\dots,\mathfrak{X}(M); \mathcal{F}(M)) \end{equation*} where all I did was change the font from calligraphic to fraktur on the first of the $r+s$ elements listed. This same typo exists in both the second and third editions. I'll assume this latter display is the correct one.]
Proof. Consider the map $\mathcal{T}^r_s(M)\to\mathfrak{T}^r_s(M)$ given by \begin{equation*} \ell(\alpha^1,\dots,\alpha^r,X_1,\dots,X_s)(m)=\ell(m)(\alpha^1(m),\dots,X_s(m)). \end{equation*} [EDIT: It seems to me that it would make more sense notationally if the right hand side $\ell$ was actually $t$.]
This map is clearly $\mathcal{F}(M)$-linear. To show it is an isomorphism, given such a multilinear map $\ell$, define $t$ by \begin{equation*} t(m)(\alpha^1(m),\dots,X_s(m))=\ell(\alpha^1,\dots,X_s)(m). \end{equation*}
To show that $t$ is well-defined we first show that, for each $v_0\in T_mM$, there is an $X\in\mathfrak{X}(M)$ such that $X(m)=v_0$, and similarly for dual vectors. Let $(U,\phi)$ be a chart at $m$ and let $T_m\varphi(v_0)=(\varphi(m),v_0')$. Define $Y\in\mathfrak{X}(U')$ by $Y(u)=(u',v_0')$ on a neighborhood $V_1$ of $\varphi(m)$, where $w=\varphi(n)$. [I have to believe there are a couple of AMR typos there. I think the $w=\varphi(n)$ was probably supposed to be $u'=\varphi(u)$, but in any case, it should not be needed, since $Y$ is supposed to be defined on $U'$, which I assume is $\varphi(U)$, so the definition of $Y$ should, I believe, read $Y(u')=(u',v_0')$.] Extend $Y$ to $U'$ so $Y$ is zero outside $V_2$, where $\mathrm{cl}(V_1)\subset V_2$, $\mathrm{cl}(V_2)\subset U'$, by means of a bump function. Define $X$ by $X_\varphi=Y$ on $U$, and $X=0$ outside $U$. Then $X(m)=v_0$. The construction is similar for dual vectors.
As in Theorem 4.2.16 [same number in both second and third editions], $\mathcal{F}(M)$-linearity of $\ell$ shows that the definition of $t(m)$ is independent of how the vectors $v_0$ (and corresponding dual vectors) are extended to fields.
The proof goes on from there to prove smoothness and to give a simplification in the finite-dimensional case.
I have two problems with this proof:
- I don't see how to show that the definition of $t$ (given $\ell$) is independent of the choice of chart $(U,\varphi)$. The referenced 4.2.16 only gives an example of how to prove that a particular definition is independent of the specifics of the bump function used to define it. It doesn't deal with independence from the choice of chart.
- I don't see how to show that $t(m)$ is continuous (multilinear). The multilinear part is easy; it's the continuous part that I can't figure out.
EDIT: It occurs to me that if this proposition is going to be true, then the meaning of $L^{r+s}_{\mathcal{F}(M)}(\mathfrak{X}^*(M),\dots,\mathfrak{X}(M);\mathcal{F}(M))$ has to be more than the stated "$\mathcal{F}(M)$-multilinear mappings". It seems that from the forward direction of the isomorphism we are going to have that \begin{equation*} |\ell(\alpha^1,\dots,X_s)(m)| =|(t(m))(\alpha^1(m),\dots,X_s(m))| \leq||t(m)||\,||\alpha^1(m)|| \cdots ||X_s(m)|| \end{equation*} so I think this condition should be part of the definition of $L^{r+s}_{\mathcal{F}(M)}(\mathfrak{X}^*(M),\dots,\mathfrak{X}(M);\mathcal{F}(M))$. That is, I think we should define \begin{equation*} \begin{split} L^{r+s}_{\mathcal{F}(M)}(&\mathfrak{X}^*(M),\dots,\mathfrak{X}(M);\mathcal{F}(M))=\\ &\{\mathcal{F}(M)\text{-multilinear }\mathcal{F}(M)\text{-valued}\,\ell: \forall m\in M,\exists c(m)\geq 0\text{ such that}\\ &|(\ell(\alpha^1,\dots,X_s))(m)|\leq c(m)||\alpha^1(m)||\cdots||X_s(m)||\}. \end{split} \end{equation*} Of course, we will have to specify, for each $m$, a norm of $T_m(M)$ and for $(T_m(M))^*$. I think this will take care of question 2. That still leaves question 1: showing chart-independence.