Pointwise - Pointwise Operations

Pointwise Operations

Examples include

\begin{align} (f+g)(x) & = f(x)+g(x) & \text{(pointwise addition)} \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text{(pointwise multiplication)} \\ (\lambda f)(x) & = \lambda \cdot f(x) & \text{(pointwise multiplication by a scalar)} \end{align}

where .

See pointwise product, scalar.

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. An example of an operation on functions which is not pointwise is convolution.

By taking some algebraic structure in the place of, we can turn the set of all functions to the carrier set of into an algebraic structure of the same type in an analogous way.

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