# Tensorial map

## Definition

Suppose is a differential manifold and are vector bundles over . A -linear map:

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

- There exists a linear map from to for every point , which gives rise to , in the sense that for any and :
- is linear with respect to the algebra of
*all*real-valued functions on . In other words, for any real-valued function we have: - is linear with respect to the algebra of all continuous real-valued functions on . In other words, for any continuous real-valued function we have the above condition.
- is linear with respect to the algebra of all smooth () functions. In other words, for any smooth real-valued function we have the above condition