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Pointwise

## Summary

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value ${\displaystyle f(x)}$ of some function ${\displaystyle f.}$ An important class of pointwise concepts are the pointwise operations, that is, 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

### Formal definition

A binary operation o: Y × YY on a set Y can be lifted pointwise to an operation O: (XY) × (XY) → (XY) on the set XY of all functions from X to Y as follows: Given two functions f1: XY and f2: XY, define the function O(f1, f2): XY by

(O(f1, f2))(x) = o(f1(x), f2(x)) for all xX.

Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity.[citation needed]

### Examples

The pointwise addition ${\displaystyle f+g}$  of two functions ${\displaystyle f}$  and ${\displaystyle g}$  with the same domain and codomain is defined by:

${\displaystyle (f+g)(x)=f(x)+g(x).}$

The pointwise product or pointwise multiplication is:

${\displaystyle (f\cdot g)(x)=f(x)\cdot g(x).}$

The pointwise product with a scalar is usually written with the scalar term first. Thus, when ${\displaystyle \lambda }$  is a scalar:

${\displaystyle (\lambda \cdot f)(x)=\lambda \cdot f(x).}$

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

### Properties

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If ${\displaystyle A}$  is some algebraic structure, the set of all functions ${\displaystyle X}$  to the carrier set of ${\displaystyle A}$  can be turned into an algebraic structure of the same type in an analogous way.

## Componentwise operations

Componentwise operations are usually defined on vectors, where vectors are elements of the set ${\displaystyle K^{n}}$  for some natural number ${\displaystyle n}$  and some field ${\displaystyle K}$ . If we denote the ${\displaystyle i}$ -th component of any vector ${\displaystyle v}$  as ${\displaystyle v_{i}}$ , then componentwise addition is ${\displaystyle (u+v)_{i}=u_{i}+v_{i}}$ .

Componentwise operations can be defined on matrices. Matrix addition, where ${\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}}$  is a componentwise operation while matrix multiplication is not.

A tuple can be regarded as a function, and a vector is a tuple. Therefore, any vector ${\displaystyle v}$  corresponds to the function ${\displaystyle f:n\to K}$  such that ${\displaystyle f(i)=v_{i}}$ , 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.[1] Using the pointwise order on functions one can concisely define other important notions, for instance:[2]

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

${\displaystyle (f_{n})_{n=1}^{\infty }}$

with
${\displaystyle f_{n}:X\longrightarrow Y}$

converges pointwise to a function ${\displaystyle f}$  if for each ${\displaystyle x}$  in ${\displaystyle X}$
${\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x).}$

## Notes

1. ^ Gierz et al., p. xxxiii
2. ^ Gierz, et al., p. 26

## References

For order theory examples:

• T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.