Definition:Pointwise Multiplication of Real-Valued Functions
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Definition
Let $f, g: S \to \R$ be real-valued functions.
Then the pointwise product of $f$ and $g$ is defined as:
- $f \times g: S \to \R:$
- $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$
where $\times$ on the right hand side denotes real multiplication.
Thus pointwise multiplication is seen to be an instance of a pointwise operation on real-valued functions.
Also denoted as
Using the other common notational forms for multiplication, this definition can also be written:
- $\forall s \in S: \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$
or:
- $\forall s \in S: \map {\paren {f g}} s := \map f s \map g s$
Also see
- Pointwise Multiplication on Real-Valued Functions is Associative
- Pointwise Multiplication on Real-Valued Functions is Commutative
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $8$