Direct Product of Vector Spaces is Vector Space

Theorem

Let $K$ be a field.

Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces.

Let $\struct {V, + , \circ}_K$ be their direct product.

Then $\struct {V, + , \circ}_K$ is a $K$-vector space.

Proof

This follows directly from Finite Direct Product of Modules 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 $26.5$