Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions

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Definition

Let $S$ be a non-empty set, and let ${\overline \R}^S$ be the set of all mappings $f: S \to \overline \R$.

Here $\overline \R$ denotes the extended set of real numbers.


Then pointwise ($\overline \R$-)scalar multiplication on ${\overline \R}^S$ is the binary operation $\cdot: \overline \R \times {\overline \R}^S \to {\overline \R}^S$ defined by:

$\forall \lambda \in \overline \R: \forall f \in {\overline \R}^S: \forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$

where the $\cdot$ on the right is extended real multiplication.


Pointwise scalar multiplication thence is an instance of a pointwise operation on extended real-valued functions.


Also defined as

Some sources only define $\cdot$ on $\R \times {\overline \R}^S$, viewing it as pointwise $\R$-scalar multiplication.

This casts $\cdot$ into the form of Pointwise Scalar Multiplication of Mappings by virtue of Extended Real Numbers form R-Algebraic Structure over Real Numbers.

However, ${\overline \R}^S$ is not a vector space over $\R$, since pointwise addition cannot be consistently defined on all of ${\overline \R}^S \times {\overline \R}^S$.


Also known as

Like the $\cdot$ for extended real multiplication, its pointwise analog (also denoted $\cdot$) is often omitted.

That is, one often encounters $\lambda f$ instead of $\lambda \cdot f$.


Furthermore, by the way pointwise $\overline \R$-scalar multiplication is defined, one often disposes of parentheses.

Thus one simply writes $\lambda \map f s$, leaving unspecified whether this means $\map {\paren {\lambda \cdot f} } s$ or $\lambda \cdot \map f s$.

This is justified as the expressions are equal in any case, and it saves one from writing excessive parentheses.


Also see