> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$, $\pi: W \to X$ be a $C^p$-[[Vector Bundle|vector bundle]] over $X$ modelled on $F$. Define > $ > W^* = \bigsqcup_{p \in X}W_p^* > $ > each fibre is the [[Topological Dual|dual]] along with $\pi^*: W^* \to X$ where $\pi(W_p^*) = \bracs{p}$. Then $\pi^*$ is also a $C^p$-vector bundle, known as the **dual bundle**. > > *Proof*. Let > $ > \tau: \pi^{-1}(U) \to U \times F \quad \tau_p \in L(W_p, F) > $ > be a trivialising map. Define > $ > \tau^*: {\pi^{*}}^{-1}(U) \to U \times F^* \quad (\tau^*)_p = (\tau_p^{-1})^* \in L(W_p^*, F^*) > $ > For any trivialising maps $\tau_1: U \to U \times F, \tau_2: U \to U \times F$, the map > $ > U \to \text{Laut}(F) \quad p \mapsto (\tau_1\circ \tau_2^{-1})_p > $ > is $C^p$. However since taking the dual reverses the composition, > $ > \begin{align*} > \braks{\tau_2^* \circ (\tau_1^*)^{-1}}_p &= \paren{\tau_2^{*}}_p \circ (\tau_1^*)_p^{-1} \\ > &= \braks{(\tau_2)_p^{-1}}^* \circ \braks{\braks{(\tau_1)_p^{-1}}^*}^{-1} \\ > &= \braks{(\tau_2)_p^{-1}}^* \circ \braks{(\tau_1)_p}^* \\ > &= [(\tau_1)_p \circ (\tau_2)_p^{-1}]^* \\ > &= [(\tau_1 \circ \tau_2^{-1})_p]^* > \end{align*} > $ > and as the map $T \mapsto T^*$ is a bounded linear map, > $ > U \to \text{Laut}(F) \quad p \mapsto (\tau_2^* \circ \tau_1^*)_p > $ > is $C^p$. > [!theorem] > > Let $X$ be a $C^p$-manifold modelled on $E$, $\pi$ be a $C^p$-vector bundle modelled on $F$, and $\pi^*$ be its dual. Let $p \in X$, $U \in \cn^o(p)$, $v \in \pi_p$ and $\phi \in \pi_p^*$, then for any trivialising map $\tau: \pi^{-1}(U) \to U \times F$, > $ > \angles{\phi, v} = \angles{(\proj_2 \circ \tau^*)\phi, (\proj_2 \circ \tau)v} > $ > *Proof*. Since $(\tau^*)_p = (\tau_p^{-1})^*$, > $ > \begin{align*} > \angles{(\proj_2 \circ \tau^*)\phi, (\proj_2 \circ \tau)v} &= \angles{(\tau^*)_p \phi, \tau_p v} \\ > &= \angles{(\tau^{-1}_p)^* \phi, \tau_p v} \\ > &= \angles{\phi \circ \tau_p^{-1}, \tau_p v} \\ > &= \angles{\phi, v} > \end{align*} > $