In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the qualifier pointwise is used to indicate that a certain property is defined by considering each value
of some function
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
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
of definition. Important
relations can also be defined pointwise.
Pointwise operations
Formal definition
A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function 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
where
.
See also
pointwise product, and
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
is some
algebraic structure, the set of all functions
to the
carrier set of
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
for some
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
and some
field . If we denote the
-th component of any vector
as
, then componentwise addition is
.
Componentwise operations can be defined on matrices. Matrix addition, where
is a componentwise operation while
matrix multiplication is not.
A
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
can be regarded as a function, and a vector is a tuple. Therefore, any vector
corresponds to the function
such that
, 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
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
on functions. With ''A'', ''B''
posets, the set of functions ''A'' → ''B'' can be ordered by ''f'' ≤ ''g'' if and only 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:
[Gierz, et al., p. 26]
* A ''
closure operator'' ''c'' on a poset ''P'' is a
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
and
idempotent self-map on ''P'' (i.e. a
projection operator
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
) 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
with
converges pointwise to a function
if for each
in
Notes
References
''For order theory examples:''
* T. S. Blyth, ''Lattices and Ordered Algebraic Structures'', Springer, 2005, .
* G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove,
D. S. Scott: ''Continuous Lattices and Domains'', Cambridge University Press, 2003.
{{PlanetMath attribution, id=7260, title=Pointwise
Mathematical terminology