Contents

Pointwise

Pointwise operations[edit]

Formal definition[edit]

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 XtoY as follows: Given two functions f₁: X → Y and f₂: X → Y, define the function O(f₁, f₂): X → Yby

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[edit]

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[edit]

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 setof${\displaystyle A}$ can be turned into an algebraic structure of the same type in an analogous way.

Componentwise operations[edit]

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.

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.

Pointwise relations[edit]

Inorder 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 ≤ gif(∀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]

• Aclosure 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 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 f if for each xinX $${\displaystyle \lim _{n\to \infty }f_{n}($$x)=f(x).}

Notes[edit]

References[edit]

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.