Pointwise

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 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. Important relations can also be defined pointwise.

Pointwise operations

Examples include

{\begin{aligned}(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{aligned}} 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.

Componentwise operations

A tuple can be regarded as a function, and a vector is a tuple. Therefore any vector $v$ corresponds to the function $f:n\to K$ such that $f(i)=v_{i}$ , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.

Pointwise relations

In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions AB can be ordered by fg if and only if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions AB with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance:

• Similarly, a projection operator k is called a kernel operator if and only if k ≤ idA.

An example of infinitary pointwise relation is pointwise convergence of functions — a sequence of functions

$\{f_{n}\}_{n=1}^{\infty }$ with

$f_{n}:X\longrightarrow Y$ $\lim _{n\rightarrow \infty }f_{n}(x)=f(x).$ 