Definition:Pointwise Operation/Real-Valued Functions

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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:

$\forall f, g \in \R^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$

In the above expression, the operator on the right hand side is the given $\oplus$ on the real numbers.


Pointwise Addition

Let $f, g: S \to \R$ be real-valued functions.


Then the pointwise sum of $f$ and $g$ is defined as:

$f + g: S \to \R:$
$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where the $+$ on the right hand side is real-number addition.


Pointwise Multiplication

Let $f, g: S \to \R$ be real-valued functions.


Then the pointwise product of $f$ and $g$ is defined as:

$f \times g: S \to \R:$
$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where $\times$ on the right hand side denotes real multiplication.



$\lambda \cdot f: S \to \R, \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$

as is done on Pointwise Scalar Multiplication


Multiary Operations

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

Let $I$ be some index set.

Let $\oplus^I: \R^I \to \R$ be an $I$-ary operation on $\R$.


Then $\oplus^I: \left[{S \to \R}\right]^I \to \left[{S \to \R}\right]$, referred to as pointwise $\oplus^I$, is defined as:

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


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


Partial Examples


Also see


Sources