pointwise operation

The canonical structure is the pointwise operations of addition and scalar multiplication.

Real-valued functions enjoy so-called pointwise operations.

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain.

The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.

Therefore any vector corresponds to the function such that , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.

Moreover, in each of these examples the sets of sections have additional algebraic structure: pointwise operations make them abelian groups, and in the examples of real and complex-valued functions the sets of sections even have a ring structure.