Definition:Pointwise Operation/Real-Valued Functions

Definition
Let $S$ be a set, and let $f,g : S \to \R$ be real-valued functions.

Let $\lambda \in \R$.

Then real-valued functions can be formed by defining (for all $s \in S$):


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

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

More generally, let $\oplus$ be a binary operation on $\R$.

Define $f \oplus g: S \to \R$, called pointwise $\oplus$, by:


 * $f \oplus g: S \to \R, \left({f \oplus g}\right) \left({s}\right) := f \left({s}\right) \oplus_\R g \left({s}\right)$

In above expressions, the subscript $\R$ of an operator expresses that its operands are real numbers.

Next, let $\left({f_i}\right)_{i \in I}, f_i: S \to \R$, be any $I$-indexed collection of real-valued functions, where $I$ is some index set.

Suppose that $\oplus^I$ is an $I$-ary operation on $\R$.

Then define $\oplus^I \left({f_i}\right)_{i \in I}: S \to \R$, called pointwise $\oplus^I$, by:


 * $\left({ \oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I_\R \left({f_i \left({s}\right) }\right)_{i \in I}$

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

Examples

 * Pointwise Addition
 * Pointwise Multiplication
 * Pointwise Scalar Multiplication
 * Absolute Value of Real-Valued Function
 * Pointwise Limit

Also see
It can be seen that these definitions instantiate the general induced operation on set of mappings.