In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value
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Pointwise operations
Examples include
where
See pointwise product, scalar.
Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. An example of an operation on functions which is not pointwise is convolution.
By taking some algebraic structure
Componentwise operations
Componentwise operations are usually defined on vectors, where vectors are elements of the set
A tuple can be regarded as a function, and a vector is a tuple. Therefore, any vector
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 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:
An example of infinitary pointwise relation is pointwise convergence of functions — a sequence of functions
with
converges pointwise to a function