Pointwise

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value ${\displaystyle f(x)}$ of some function ${\displaystyle f.}$ 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

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]}

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.

Pointwise operations inherit such properties as

. If ${\displaystyle A}$ is some

${\displaystyle X}$

${\displaystyle A}$

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.

In

• 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 x in X $${\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x).}$$

Notes

1. ^ Gierz et al., p. xxxiii
2. ^ Gierz, et al., p. 26

References

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.

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