Definition:Vector Space of All Mappings
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Definition
Let $\struct {K, +, \circ}$ be a division ring.
Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space.
Let $S$ be a set.
Let $G^S$ be the set of all mappings from $S$ to $G$.
Then $\struct {G^S, +_G', \circ}_K$ is a $K$-vector space, where:
- $+_G'$ is the operation induced on $G^S$ by $+_G$
- $\forall \lambda \in K: \forall f \in G^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$
This is the $K$-vector space $G^S$ of all mappings from $S$ to $G$.
Examples
The most important case of this example is when $\struct {G^S, +_G', \circ}_K$ is the $K$-vector space $\struct {K^S, +_K', \circ}_K$.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.4$