Pointwise

Componentwise operations

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.

A tuple 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

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 (∀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. 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 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 $${\displaystyle (f_n{)}_{n=1}^{\mathrm{\infty }}}$$ with $${\displaystyle f_n:X⟶Y}$$ converges pointwise to a function f if for each x in X $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=f(x).}$$