Definition:Pointwise Scalar Multiplication of Number-Valued Function

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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 $f_\lambda: S \to \mathbb F: f_\lambda \left({s}\right) = \lambda$, the following definition arises:


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

$\times: \mathbb F \times \mathbb F^S \to \mathbb F^S: \forall \lambda \in \mathbb F, f \in \mathbb F^S:$
$\forall s \in S: \left({\lambda \times f}\right) \left({s}\right) := \lambda \times f \left({s}\right)$

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


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


Also denoted as

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

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

or:

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


Specific Number Sets

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

Integer-Valued Function

Let $f: S \to \Z$ be an integer-valued function.

Let $\lambda \in \Z$ be an integer.


Then the pointwise scalar product of $f$ by $\lambda$ is defined as:

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

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


Rational-Valued Function

Let $f: S \to \Q$ be an rational-valued function.

Let $\lambda \in \Q$ be an rational number.


Then the pointwise scalar product of $f$ by $\lambda$ is defined as:

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

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


Real-Valued Function

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.


Complex-Valued Function

Let $f: S \to \C$ be an complex-valued function.

Let $\lambda \in \C$ be an complex number.


Then the pointwise scalar product of $f$ by $\lambda$ is defined as:

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

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


Also see