Direct Product of Vector Spaces is Vector Space

Theorem
Let $$G_1, G_2, \ldots, G_n$$ be $K$-vector spaces.

Let:
 * $$G = \prod_{k=1}^n G_k$$

Then $$\left({G, +, \circ}\right)_K$$ is a $K$-vector space where:


 * $$+$$ is the operation induced on $G$ by the operations $$+_1, +_2, \ldots, +_n$$ on $$G_1, G_2, \ldots, G_n$$


 * $$\circ$$ is defined as $$\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}\right)$$

Also see

 * Module Product

Proof
This follows directly from Module Product and the definition of vector space.