Definition:Pointwise Addition
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:
- $f + g: S \to \Z:$
- $\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$
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:
- $f + g: S \to \Q:$
- $\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$
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:
- $f + g: S \to \C:$
- $\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$
where $+$ on the right hand side 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 pointwise 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 pointwise 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
- Definition:Pointwise Operation on Number-Valued Functions: a more general concept of which these definitions can be seen to be instantiations.