> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]], then $\vf(X)$ is the set of all $C^{p - 1}$-[[Vector Field|vector fields]] on $X$. > [!theorem] > > Let $X$ be a Hausdorff [[Manifold|manifold]] of class $C^p$, and $\alpha, \beta$ be [[Vector Field|vector fields]] of class $C^{p - 1}$. Since $\alpha(p), \beta(p) \in T_pX$, define > $ > (\lambda \alpha + \beta)(p) = \lambda \alpha(p) + \beta (p) > $ > then $\lambda \alpha + \beta$ is also a vector field of class $C^{p - 1}$, and $\vf(X)$ is a $\real$-[[Vector Space|vector space]]. > > *Proof*. Let $p \in X$ and $(U, \varphi)$ be a [[Atlas|chart]] at $p$, which induces a bundle chart > $ > \tilde \varphi: \pi^{-1}(U) \to E \times E \quad (p, v) \mapsto (\hat p, \hat v) > $ > For fixed $q$, the map $\tilde \varphi_q: \pi^{-1}(q) \to E$ is a [[Space of Toplinear Isomorphisms|toplinear isomorphism]], so addition and scalar multiplication has coordinate representation > $ > (\lambda \alpha + \beta)_{U, \pi^{-1}(U)} = \lambda \alpha_{U, \pi^{-1}(U)} + \beta_{U, \pi^{-1}(U)} > $ > meaning that $\lambda \alpha + \beta$ is $C^{p - 1}$ at $p$. Hence $\lambda \alpha + \beta$ is a vector field of class $C^{p - 1}$. > [!theorem] > > Let $\xi: X \to TX$ be a vector field of class $C^{p - 1}$ and let $f \in C^{p - 1}(X)$. Define > $ > f \xi(p) = f \cdot \xi(p) > $ > then $f \xi$ is a vector field of class $C^{p - 1}$, and $\vf(X)$ is a $C^{p - 1}(X)$-module. > > *Proof*. Similarly, $f \xi$ has coordinate representation with respect to $(U, \varphi)$ and its bundle chart > $ > (f \xi)_{U, \pi^{-1}(U)} = (f \circ \varphi^{-1}) \cdot \xi_{U, \pi^{-1}(U)} > $ > which is of class $C^{p - 1}$. Therefore $f \xi \in \vf(X)$.