# Tensorial map

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## Definition

Suppose $M$ is a differential manifold and $E,F$ are vector bundles over $M$. A $\R$-linear map: $f: \Gamma(E) \to \Gamma(F)$

between the spaces of sections is termed tensorial or pointwise if it satisfies the following equivalent conditions:

1. There exists a linear map $f_m$ from $E_m$ to $F_m$ for every point $m$, which gives rise to $f$, in the sense that for any $m \in M$ and $s \in \Gamma(E)$: $f_m(s(m)) = (f(s))(m)$

1. $f$ is linear with respect to the algebra of all real-valued functions on $M$. In other words, for any real-valued function $g: M \to \R$ we have: $g(m)(f(s))(m) = f(g(m)s)(m)$

1. $f$ is linear with respect to the algebra of all continuous real-valued functions on $M$. In other words, for any continuous real-valued function $g: M \to \R$ we have the above condition.
2. $f$ is linear with respect to the algebra of all smooth ( $C^\infty$) functions. In other words, for any smooth real-valued function $g: M \to \R$ we have the above condition