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 $(V,+,\circ)_K$ be their direct product.

Then $(V,+,\circ)_K$ is a $K$-vector space.

Proof
This follows directly from Finite Direct Product of Modules is Module and the definition of vector space.