Definition:Pointwise Scalar Multiplication of Real-Valued Function

Definition
Let $S$ be a non-empty set. Let $f: S \to \R$ be an real-valued function.

Let $\lambda \in \R$ be an real number.

Then the pointwise scalar product of $f$ by $\lambda$ is defined as:
 * $\lambda \times f: S \to \R:$
 * $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$

where the $\times $ on the is real multiplication.

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

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

Also see

 * Definition:Pointwise Addition of Real-Valued Functions
 * Definition:Pointwise Multiplication of Real-Valued Functions


 * Definition:Pointwise Scalar Multiplication of Number-Valued Function: a more general concept of which this is a specific instance