Definition:Pointwise Operation/Real-Valued Functions

Definition
Let $S$ be a non-empty set. Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers.

Let $\oplus$ be a binary operation on $\R$.

Define $\oplus: \R^S \times \R^S \to \R^S$, called pointwise $\oplus$, by:


 * $\forall f, g \in \R^S: \forall s \in S: \left({f \oplus g}\right) \left({s}\right) := f \left({s}\right) \oplus g \left({s}\right)$

In the above expression, the operator on the RHS is the given $\oplus$ on the real numbers.

Specific instances of this concept are as follows:

Pointwise Multiplication

 * $\lambda \cdot f: S \to \R, \left({\lambda \cdot f}\right) \left({s}\right) := \lambda \cdot f \left({s}\right)$
 * $f + g: S \to \R, \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$
 * $f \cdot g: S \to \R, \left({f \cdot g}\right) \left({s}\right) := f \left({s}\right) \cdot g \left({s}\right)$

as is done on Pointwise Scalar Multiplication, Pointwise Addition and Pointwise Multiplication, respectively.

Multiary Operations
Next, for ease of notation, write $\left[{S \to \R}\right]$ for $\R^S$.

Let $I$ be some index set, and suppose that $\oplus^I: \R^I \to \R$ is an $I$-ary operation on $\R$.

Then define $\oplus^I: \left[{S \to \R}\right]^I \to \left[{S \to \R}\right]$, called pointwise $\oplus^I$, by:


 * $\forall \left({f_i}\right)_{i \in I} \in \left[{S \to \R}\right]^I: \forall s \in S: \left({ \oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I \left({f_i \left({s}\right) }\right)_{i \in I}$

Also known as
When $\oplus$ or $\oplus^I$ has a distinguished name, it is usual to name the corresponding pointwise operation by prepending pointwise to that name.

Also defined as
Sometimes an operation cannot be consistently defined on all of $\R^I$. Often one then still speaks about a pointwise operation by suitably restricting above definition, adapting it wherever necessary.

Examples of such suitably restricted pointwise operations are listed under Partial Examples below.

Examples

 * Pointwise Addition
 * Pointwise Multiplication
 * Pointwise Scalar Multiplication
 * Pointwise Maximum
 * Pointwise Minimum
 * Absolute Value

Partial Examples

 * Pointwise Limit

Also see
It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.