> [!theorem] > > Let $E$ and $F$ be [[Banach Space|Banach spaces]], and $T: E \to F$ be a [[Bounded Linear Map|bounded linear map]]. Define a mapping > $ > \symbi(T): \symbi(F) \to \symbi(E) \quad \lambda(\cdot, \cdot ) \mapsto \lambda(T \cdot, T\cdot) > $ > then $\symbi$ is a [[Differentiable Functor of One Variable|differentiable functor]]. If $X$ is a $C^p$-[[Manifold|manifold]] and $\pi$ is a $C^{p - 1}$ [[Vector Bundle|vector bundle]], then this functor induces a $C^{p - 1}$-vector bundle. > > A [[Sections of Vector Bundles|section of]] $\symbi(\pi)$ is a **symmetric bilinear form** on $\pi$. > [!theorem] > > Let $g \in \symbi(\pi)$ be a symmetric bilinear form on $\pi$, and $\tau: \pi^{-1}(U) \to U \times E$ be a local trivialisation, then $g$ has **local representation**, > $ > \begin{CD} > U @>{g}>> \pi^{-1}(U) @>{\tau}>> U \times \symbi(E) @>{\proj_2}>> \symbi(E) > \end{CD} > $ > as a $C^{p - 1}$-morphism $\wh{g}: U \to \symbi(E)$. If each $\wh g_x$ is associated with a linear operator $A_x$, then the map $U \to L(E, E)$ is also a morphism. > [!definition] > > Let $g \in \symbi$ be a symmetric bilinear form on $\pi$. For any $p \in X$ and $x, y \in \pi^{-1}(p)$, denote > $ > \angles{x, y}_p = g_p(x, y) > $