Vector Space of All Mappings is Vector Space

Theorem

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 $\struct {G^S, +_G', \circ}_R$ be the vector space of all mappings from $S$ to $G$.

Then $\struct {G^S, +_G', \circ}_K$ is a $K$-vector space.

Proof

Follows directly from Module of All Mappings is Module and the definition of vector space.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.4$