# Pointwise

In mathematics, the qualifier **pointwise** is used to indicate that a certain property is defined by considering each value of some function 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.

## Contents

## Pointwise operations

Examples include

where .

See pointwise product, scalar.

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.

By taking some algebraic structure in the place of , we can turn the set of all functions to the carrier set of 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 for some natural number and some field . can be generalized to a set. If we denote the -th component of any vector as , then componentwise addition is .

A tuple can be regarded as a function, and a vector is a tuple. Therefore any vector corresponds to the function such that , 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 *A* → *B* can be ordered by *f* ≤ *g* 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 *A* → *B* with pointwise order.^{1} Using the pointwise order on functions one can concisely define other important notions, for instance:^{2}

- A closure operator
*c*on a poset*P*is a monotone and idempotent self-map on*P*(i.e. a projection operator) with the additional property that id_{A}≤*c*, where id is the identity function.

- Similarly, a projection operator
*k*is called a kernel operator if and only if*k*≤ id_{A}.

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

with

converges pointwise to a function if for each in

## Notes

## 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.

*This article incorporates material from Pointwise on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

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