Definition:Pointwise Scalar Multiplication of Complex-Valued Function
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Definition
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.
This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping:
- $f_\lambda: S \to \C: \map {f_\lambda} s = \lambda$
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 Complex-Valued Functions
- Definition:Pointwise Multiplication of Complex-Valued Functions
- Definition:Pointwise Scalar Multiplication of Number-Valued Function: a more general concept of which this is a specific instance