Definition:Direct Product of Vector Spaces/General Case

Definition
Let $K$ be a field.

Let $\left\{ \left\langle V_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$ be a family of $K$-vector spaces.

The (external) direct product of $\left\{ \left\langle V_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$ is their module direct product.