Definition:Support of Mapping to Algebraic Structure

From ProofWiki
Jump to navigation Jump to search

Definition

Real-Valued Function on an Abstract Set

Let $S$ be a set.

Let $f: S \to \R$ be a real-valued function.


The support of $f$ is the set of elements $x$ of $S$ whose values under $f$ are non-zero.

That is:

$\operatorname{supp} \left({f}\right) := \left\{{x \in S: f \left({x}\right) \ne 0}\right\}$

General Real-Valued Function in $\R^n$

General Algebraic Structure

Let $(A, *)$ be an algebraic structure with an identity element $e$.

Let $S$ be a set.

Let $f : S \to A$ be a mapping.


The support of $f$ is the set:

$\operatorname{supp}(f) = \{s \in S : f(s) \neq e\}$

Sequence

Note that by definition, a sequence is a mapping, so that the definition of support applies in particular to sequences.


Also see