# Definition:Pointwise Addition

## Contents

## Definition

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:

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

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.