# 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

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