For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.
Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their purely abstract incarnation (monoidal categories). To make the intent clear, this tag should only be accompanied by relevant other tags specifying the context.
If $V_1$ and $V_2$ are vector spaces over some field $F$, then the tensor product of $V_1$ and $V_2$ is a vector space $V_1\otimes V_2$ for which there is a bilinear map $\beta\colon V_1\times V_2\longrightarrow V_1\otimes V_2$ with the following property: for each bilinear map $B$ from $V_1\times V_2$ into a vector space $W$, there is one and only one linear map $f_B\colon V_1\otimes V_2\longrightarrow W$ such that $B=f_B\circ\beta$. If $v_1\in V_1$ and $v_2\in V_2$, then $\beta(v_1,v_2)$ is usually denoted by $v_1\otimes v_2$.
If $V_1$ and $V_2$ are finite-dimensional, then $V_1\otimes V_2$ is also finite-dimensional and $\dim(V_1\otimes V_2)=\dim(V_1)\cdot\dim(V_2)$.