\begin{definition}[Definition:Direct Product of Vector Spaces/Finite Case]
Let $K$ be a field.
Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces.
Let:
:$\ds V = \prod_{k \mathop = 1}^n V_k$
be their cartesian product.
Let:
:$+$ be the operation induced on $V$ by the operations $+_1, +_2, \ldots, +_n$ on $V_1, V_2, \ldots, V_n$
:$\circ$ be defined as $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} := \tuple {\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}$
$\struct {V, +, \circ}_K$ is called the '''direct product of $V_1, \ldots, V_n$'''.
\end{definition}