Definition:Pointwise Addition

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Definition

Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$.


The (binary) operation of pointwise addition is defined on $\mathbb F^S$ as:

$+: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where the $+$ on the right hand side is conventional arithmetic addition.


Specific Number Sets

Specific instantiations of this concept to particular number sets are as follows:


Integer-Valued Functions

Let $f, g: S \to \Z$ be integer-valued functions.


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

$f + g: S \to \Z:$
$\forall s \in S: \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$

where the $+$ on the right hand side is integer addition.


Rational-Valued Functions

Let $f, g: S \to \Q$ be rational-valued functions.


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

$f + g: S \to \Q:$
$\forall s \in S: \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$

where the $+$ on the right hand side is integer addition.


Real-Valued Functions

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


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

$f + g: S \to \R:$
$\forall s \in S: \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$

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


Complex-Valued Functions

Let $f, g: S \to \C$ be complex-valued functions.


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

$f + g: S \to \C:$
$\forall s \in S: \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$

where the $+$ on the RHS is complex addition.


Pointwise Addition on Ring of Mappings

Let $\struct {R, +, \circ}$ be a ring.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$.


The operation $+’$ induced by $+$ on the ring of mappings from $S$ to $R$ is called pointwise addition and is defined as:

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


Pointwise Addition on Ring of Sequences

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$.


The operation $+’$ induced by $+$ on the ring of sequences is called pointwise addition and is defined as:

$\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$


Also see


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