Definition:Pointwise Scalar Multiplication of Real-Valued Function

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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:

$\lambda \times f: S \to \R:$
$\forall s \in S: \left({\lambda \times f}\right) \left({s}\right) := \lambda \times f \left({s}\right)$

where the $\times $ on the right hand side is real multiplication.

This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping:

$f_\lambda: S \to \R: f_\lambda \left({s}\right) = \lambda$

Also denoted as

Using the other common notational forms for multiplication, this definition can also be written:

$\forall s \in S: \left({\lambda \cdot f}\right) \left({s}\right) := \lambda \cdot f \left({s}\right)$


$\forall s \in S: \left({\lambda f}\right) \left({s}\right) := \lambda f \left({s}\right)$

Also see

This is a specific instance of a Pointwise Scalar Multiplication of Number-Valued Function.