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 $$\left({G, +: \circ}\right)_K$$ us a $K$-vector space where:
 * $$+$$ is the operation induced on $$G$$ by the operations on $$G_1, G_2, \ldots, G_n$$;
 * $$\circ$$ is defined as $$\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}\right)$$.

Compare Module Product.