Category:Definitions/Pointwise Operations
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This category contains definitions related to Pointwise Operations.
Related results can be found in Category:Pointwise Operations.
Let $S$ be a set.
Let $\struct {T, \circ}$ be an algebraic structure.
Let $T^S$ be the set of all mappings from $S$ to $T$.
Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.
Then the operation $f \oplus g$ is defined on $T^S$ as follows:
- $f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$
The operation $\oplus$ is called the pointwise operation on $T^S$ induced by $\circ$.
Induced Structure
The algebraic structure $\struct {T^S, \oplus}$ is called the algebraic structure on $T^S$ induced by $\circ$.
Subcategories
This category has the following 3 subcategories, out of 3 total.
P
- Definitions/Pointwise Addition (14 P)
Pages in category "Definitions/Pointwise Operations"
The following 25 pages are in this category, out of 25 total.
P
- Definition:Pointwise Addition
- Definition:Pointwise Addition of Mappings
- Definition:Pointwise Inverse
- Definition:Pointwise Maximum of Mappings
- Definition:Pointwise Maximum of Mappings/Real-Valued Functions
- Definition:Pointwise Minimum of Mappings
- Definition:Pointwise Multiplication
- Definition:Pointwise Operation
- Definition:Pointwise Operation on Complex-Valued Functions
- Definition:Pointwise Operation on Extended Real-Valued Functions
- Definition:Pointwise Operation on Integer-Valued Functions
- Definition:Pointwise Operation on Number-Valued Functions
- Definition:Pointwise Operation on Rational-Valued Functions
- Definition:Pointwise Operation on Real-Valued Functions
- Definition:Pointwise Operation on Real-Valued Functions/Multiary
- Definition:Pointwise Operation/Induced Structure
- Definition:Pointwise Operation/Real-Valued Functions
- Definition:Pointwise Scalar Multiplication
- Definition:Pointwise Scalar Multiplication of Mappings
- Definition:Pointwise Supremum
- Definition:Pointwise Supremum of Extended Real-Valued Functions
- Definition:Pointwise Supremum of Real-Valued Functions