Definition:Pointwise Scalar Multiplication of Number-Valued Function

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 $f: S \to \mathbb F$.

When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$, the following definition arises:

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


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

where the $\times$ on the is conventional arithmetic multiplication.

This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$

Also denoted as
Using the other common notational forms for multiplication, this definition can also be written:
 * $\forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$

or:
 * $\forall s \in S: \map {\paren {\lambda f} } s := \lambda \map f s$

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

Also see

 * Definition:Pointwise Addition
 * Definition:Pointwise Multiplication