Inmathematics, the qualifierpointwise is used to indicate that a certain property is defined by considering each value${\displaystyle f(x)}$ of somefunction${\displaystyle f.}$ An important class of pointwise concepts are thepointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in thedomain of definition. Importantrelations can also be defined pointwise.

A binary operationo:Y ×Y →Y on a setY can be lifted pointwise to an operationO: (X→Y) × (X→Y) → (X→Y) on the setX →Y of all functions fromX toY as follows: Given two functionsf₁:X →Y andf₂:X →Y, define the functionO(f₁,f₂):X →Y by

Commonly,o andO are denoted by the same symbol. A similar definition is used for unary operationso, and for operations of otherarity.^{[citation needed]}

The pointwise addition${\displaystyle f+g}$ of two functions${\displaystyle f}$ and${\displaystyle g}$ with the same domain andcodomain 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 ascalar:

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

An example of an operation on functions which isnot pointwise isconvolution.

Pointwise operations inherit such properties asassociativity,commutativity anddistributivity from corresponding operations on thecodomain. If${\displaystyle A}$ is somealgebraic structure, the set of all functions${\displaystyle X}$ to thecarrier set of${\displaystyle A}$ can be turned into an algebraic structure of the same type in an analogous way.

Componentwise operations are usually defined on vectors, where vectors are elements of the set${\displaystyle K^n}$ for somenatural number${\displaystyle n}$ and somefield${\displaystyle K}$. 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 whilematrix multiplication is not.

Atuple 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.

Inorder theory it is common to define a pointwisepartial order on functions. WithA,Bposets, the set of functionsA →B can be ordered by definingf ≤g if(∀x ∈ A)f(x) ≤g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B arecontinuous lattices, then so is the set of functionsA →B with pointwise order.^[1] Using the pointwise order on functions one can concisely define other important notions, for instance:^[2]

• Aclosure operatorc on a posetP is amonotone andidempotent self-map onP (i.e. aprojection operator) with the additional property that id_A ≤c, where id is theidentity function.
• Similarly, a projection operatork is called akernel operator if and only ifk ≤ id_A.

An example of aninfinitary pointwise relation ispointwise convergence of functions—asequence of functions$${\displaystyle (f_n{)}_{n=1}^{\mathrm{\infty }}}$$with$${\displaystyle f_n:X⟶Y}$$converges pointwise to a functionf if for eachx inX$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=f(x).}$$

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 onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

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