Definition:Pointwise Scalar Multiplication of Number-Valued Function
Definition
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 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 $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$
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:
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:
- $\lambda \times f: S \to \Z:$
- $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
where $\times$ 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:
- $\lambda \times f: S \to \Q:$
- $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
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:
- $\lambda \times f: S \to \R:$
- $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
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:
- $\lambda \times f: S \to \C:$
- $\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
where $\times$ on the right hand side denotes complex multiplication.