Let $V$ and $W$ be vector spaces over a field $k$. Let $F(V \times W)$ denote the free vector space on the set $V \times W$: this is the vector space with basis $\{e_{(v,w)} : v \in V, w \in W\}$, consisting of all formal finite linear combinations of pairs. Let $R$ be the subspace of $F(V \times W)$ generated by all elements of the following forms:
\begin{align}
e_{(v_1 + v_2, w)} - e_{(v_1, w)} - e_{(v_2, w)}, \\
e_{(v, w_1 + w_2)} - e_{(v, w_1)} - e_{(v, w_2)}, \\
e_{(\lambda v, w)} - \lambda e_{(v, w)}, \\
e_{(v, \lambda w)} - \lambda e_{(v, w)},
\end{align}
for all $v, v_1, v_2 \in V$, $w, w_1, w_2 \in W$, and $\lambda \in k$. The tensor product of $V$ and $W$ over $k$ is the quotient vector space
\begin{align}
V \otimes_k W := F(V \times W) / R.
\end{align}
The image of $e_{(v,w)}$ in the quotient is written $v \otimes w$ and called an elementary tensor. The canonical map
\begin{align}
\otimes: V \times W &\to V \otimes_k W \\
(v, w) &\mapsto v \otimes w
\end{align}
is bilinear by construction (the relations in $R$ encode precisely bilinearity). When the field $k$ is understood, we write $V \otimes W$ for $V \otimes_k W$.
