Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value

f

(

x

)

{\displaystyle f(x)}

of some function

f

.

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

The pointwise addition

f

+

g

{\displaystyle f+g}

of two functions

f

{\displaystyle f}

and

g

{\displaystyle g}

with the same domain and codomain is defined by:

The pointwise product or pointwise multiplication is:

The pointwise product with a scalar is usually written with the scalar term first. Thus, when

λ

{\displaystyle \lambda }

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.

If

A

{\displaystyle A}

is some algebraic structure, the set of all functions

X

{\displaystyle X}

to the carrier set of

A

{\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

K

n

{\displaystyle K^{n}}

for some natural number

n

{\displaystyle n}

and some field

K

{\displaystyle K}

.