Pointwise

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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\}_{n=1}^\infty

with

f_n:X \longrightarrow Y

converges pointwise to a function f if for each x in X

\lim_{n \rightarrow \infty} 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

(f+g)(x)\, = f(x)+g(x)\, (pointwise addition)
(f\cdot g)(x)\, = f(x) \cdot g(x)\, (pointwise product)
(\lambda f)(x)\, = \lambda \cdot f(x)\, (pointwise multiplication by a scalar)


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

An example of an operation on functions which is not pointwise is convolution.

This article incorporates material from Pointwise on PlanetMath, which is licensed under the GFDL.

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