> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]] modelled on $E$. Let > $ > T^*X = \bigsqcup_{p \in X}T_p^*X > $ > be the disjoint union of all [[Tangent Space|tangent spaces]] at each point and $\pi: T^*X \to X$ be the canonical projection. Let $(U, \varphi)$ be a chart for $X$, and define > $ > \tau_U: \pi^{-1}(U) \to U \times E^* \quad (p, \omega ) \mapsto (p, \wh \omega) > $ > where we identify $(T_{(p, U, \varphi)}X)^*$ with $E^*$. Then the collection of all such maps forms a trivialising cover for $T^*X$, making it a [[Vector Bundle|vector bundle]] of class $C^{p - 1}$, known as the **cotangent bundle**. > > *Proof*. Let $(U, \varphi)$ and $(V, \psi)$ be charts with intersecting domains, $\tau_U$ and $\tau_V$ be the corresponding trivialising maps, and $p \in U \cap V$. > > Since $D(\psi \circ \varphi^{-1})(p)$ is a [[Space of Toplinear Isomorphisms|toplinear isomorphism]], so is its dual. So the translation between > $ > (\tau_{V} \circ \tau_{U}^{-1})_p = D(\psi \circ \varphi^{-1})(p)^*: (T_{(p, V, \psi)}X)^* \to (T_{(p, U, \varphi)}X)^* > $ > is also a toplinear isomorphism. Moreover, since the map from a linear map to its dual is linear and norm-preserving it is smooth. The map $p \mapsto (\tau_V \circ \tau_U^{-1})_p$ has representation > $ > \begin{CD} > p @>{D(\psi \circ \varphi^{-1})}>> \text{Laut}(E) @>{\text{dual}}>> \text{Laut}(E^*) > \end{CD} > $ > as a composition between a $C^{p-1}$ map and a smooth map, making it $C^{p - 1}$. Therefore $\tau_V$ and $\tau_U$ are VB-equivalent. > > As the charts cover $X$, the collection of all such trivialising maps form a trivialising cover, giving $X$ a unique manifold and vector bundle structure.