> [!definition] > > Let $\catc, \mathfrak{D}$ be subcategories of [[Banach Space|Banach spaces]], $E_1, E_2 \in \catc$, and $\lambda: \catc \to \mathfrak{D}$ be a [[Differentiable Functor|differentiable functor]]. Let $X$ and $Y$ be $C^p$-[[Manifold|manifolds]] modelled on the [[Banach Space|Banach spaces]] $E_1$ and $E_2$ respectively, and $F: X \to Y$ be a $C^{p}$-[[Manifold Morphism|morphism]], then it induces a bounded linear map on each fibre: > $ > \lambda(dF_p): \lambda(T_{F(p)}Y) \to \lambda(T_pX) > $ > known as the **pointwise pullback** by $F$ at $p$. If $\omega: Y \to \lambda (TY)$ is a [[Sections of Vector Bundles|section of]] $\lambda(TY)$, then there exists a section of $\lambda(TX)$, > $ > \begin{CD} > X @>F>> Y \\ > @V{F_*\omega}VV @VV{\omega}V \\ > \lambda(TX) @<<\lambda(dF)< \lambda(TY) > \end{CD} > $ > defined by > $ > F_*\omega: X \to \lambda (TX) \quad p \mapsto \angles{\lambda(dF_p), \omega_{F(p)}} > $ > > *Proof*. Composite of a bounded bilinear map and differentiable maps. > [!theorem] > > Let $\catc, \mathfrak D$ be subcategories of Banach spaces, $X$ be a $C^p$-manifold modelled on $E \in \catc$, $\lambda: \catc \to \mathfrak D$ be a differentiable functor, and $\omega \in \mathbf{S}(\lambda(TX))$ be a section. > > If $(U, \varphi)$ is a chart, then the local representation > $ > \begin{CD} > \wh U @>{\varphi^{-1}}>> U @>{\omega}>> \lambda(TU) @>{\wh \varphi}>> U \times \lambda(E) @>{\proj_2}>> \lambda(E) > \end{CD} > $ > is the pullback of $\omega$ to $\lambda(T \wh U)$ by $\varphi^{-1}$, where $\wh \varphi$ is the bundle chart created by the functor. > > *Proof*. Firstly, the map $\varphi^{-1}: \wh U \to U$ has differential corresponding to the identification > $ > d\varphi^{-1}(p): E = T_{(U, \varphi, p)}U \to T_pU > $ > representing the isomorphism between the [[Concrete Tangent Space|concrete tangent spaces]] and the [[Tangent Space|tangent spaces]]. While the concrete tangent space is not actually equal to the model space, we make the canonical identification here since we are working with the space to begin with. The bundle chart then has the representation > $ > \wh \varphi: TU \to U \times \lambda(E) \quad (p, v) \mapsto (p, \lambda(d\varphi^{-1}(p))(v)) > $ > where > $ > \lambda(d\varphi^{-1}(p)): T_pU \to \lambda(T_{(U, \varphi, p)}) = \lambda(E) > $ > thus $\proj_2 \circ \wh \varphi = \lambda(d\varphi^{-1})$, and the local representation corresponds exactly to the pullback.