In mathematics, the qualifier pointwise of some function 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 on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by 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.

Examples

with the same domain and codomain is defined by: The pointwise product or pointwise multiplication is: is a scalar:

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 is some algebraic structure to the carrier set can be turned into an algebraic structure of the same type in an analogous way.

Componentwise operations

for some natural number and some field.

is a componentwise operation while matrix multiplication is not.

A tuple, 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 defining f ≤ g if . 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. Using the pointwise order on functions one can concisely define other important notions, for instance:

• 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 idA ≤ c, where id is the identity function.

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

An example of an infinitary pointwise relation is pointwise convergence of functionsa sequence 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, .

• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.