AskDefine | Define pointwise

Extensive Definition

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An example is pointwise convergence of functions — a sequence of functions
f_n:X \longrightarrow Y
converges pointwise to a function f if for each x in X
\lim_ f_n(x) = f(x).
An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values separately for each point in the domain of definition. These include
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.
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