---- > [!definition] Definition. ([[dual vector bundle]]) > The **dual vector bundle** of a [[vector bundle]] $\pi: E \to X$ is a (the?) vector bundle $\pi^{*}:E^{*} \to X$ obtained by passing fiberwise to the [[dual vector space]], i.e., the [[Hom bundle]] $\text{Hom}(E, \mathbb{R} \times X)$. > Specifically, $\pi^{*}$ is constructed by starting with a trivializing cover $\{ U_{\alpha} \}_{\alpha\in I}$ of $E$, giving rise to [[transition functions for a vector bundle over a smooth manifold|transition functions]] $\{\psi_{\alpha \beta}\}_{\alpha \in I, \beta \in J}$, and applying the [[the Steenrod construction of a vector bundle over a smooth manifold|Steenrod construction]] to $(\{ U_{\alpha} \}_{\alpha \in I}, \{ \psi_{\beta \alpha}^{*} \}_{\alpha \in I, \beta \in J})$. ---- #### Let's apply the [[the Steenrod construction of a vector bundle over a smooth manifold|Steenrod construction]] as prescribed. Note that $\psi_{\beta \alpha}=\psi_{\alpha \beta ^{-1}}$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```