Pointwise

(lower plot, blue) and ln (red). The highlighted vertical slice shows the computation at the point x=2π. 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 The pointwise addition f+g of two functions f and g with the same domain and codomain is defined by: The pointwise product or pointwise multiplication is: The pointwise product with a scalar is usually written with the scalar term first. Thus, when \lambda is a 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 A is some algebraic structure, the set of all functions X to the carrier set of A 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 K^n for some natural number n and some field K. If we denote the i-th component of any vector v as v_i, then componentwise addition is (u+v)_i = u_i+v_i. Componentwise operations can be defined on matrices. Matrix addition, where (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 v corresponds to the function f:n\to K such that 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 . 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 idA ≤ c, where id is the identity function. • Similarly, a projection operator k is called a kernel operator if and only if k ≤ idA. An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions (f_n)_{n=1}^\infty with f_n:X \longrightarrow Y converges pointwise to a function if for each in \lim_{n \to \infty} f_n(x) = f(x). == Notes ==