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 is the given $\oplus$ on the real numbers.

Specific Instantiations
When $\oplus$ has a specific name, it is usual to name the corresponding pointwise operation by prepending pointwise to that name:

Pointwise Multiplication

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

as is done on Pointwise Scalar Multiplication

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

 * Definition:Pointwise Addition of Real-Valued Functions
 * Definition:Pointwise Multiplication of Real-Valued Functions
 * Definition:Pointwise Scalar Multiplication of Real-Valued Functions
 * Definition:Pointwise Maximum of Real-Valued Functions
 * Definition:Pointwise Minimum of Real-Valued Functions
 * Definition:Absolute Value of Real-Valued Function

Partial Examples

 * Definition:Pointwise Limit of Real-Valued Functions

Also see

 * Definition:Pointwise Operation on Number-Valued Functions: a more general concept