Definition:Pointwise Addition of Mappings

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Definition

Let $S$ be a non-empty set.

Let $\struct {G, \circ}$ be a commutative semigroup.

Let $G^S$ be the set of all mappings from $S$ to $G$.

Then pointwise addition on $G^S$ is the binary operation $\circ: G^S \times G^S \to G^S$ (the $\circ$ is the same as for $G$) defined by:

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

The double use of $\circ$ is justified as $\struct {G^S, \circ}$ inherits all abstract-algebraic properties $\struct {G, \circ}$ might have.

This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure.

Pointwise Multiplication

Let $\circ$ be used with multiplicative notation.

Then the operation defined above is called pointwise multiplication instead.

Examples

• Definition:Pointwise Addition of Real-Valued Functions
• Definition:Pointwise Addition of Extended Real-Valued Functions
• Definition:Pointwise Multiplication of Real-Valued Functions
• Definition:Pointwise Multiplication of Extended Real-Valued Functions

Also see

• Definition:Pointwise Scalar Multiplication of Mappings, a similar concept commonly used with maps on vector spaces.