[Math] Proof of Universal Mapping Property for tensor product of vector spaces

Question

Let $V$ and $W$ be two vector spaces. We define the tensor product of $V$ and $W$, denoted by $V \otimes W$, like wikipedia does that (https://en.wikipedia.org/wiki/Tensor_product). I want to prove the UMP. It is: if
$$
\begin{array}{llccl}
\pi & : & V \times W & \to & V \otimes W
\\
& & (v , w) & \mapsto & v \otimes w\mbox{,}
\end{array}
$$
$U$ is a vector space on $K$ and $l : V \times W \to U$ is a bilinear map, then there exists a unique linear map $\tilde{l} : V \otimes W \to U$ such that $\tilde{l} \circ \pi = l$ on $V \times W$. Thank you very much in advance.

Answer 1

For existence define $\tilde{l}(v\otimes w):=l(v,w)$ and you extend this linearly. This means you furthermore define $\tilde{l}(a\otimes b+c\otimes d):=\tilde{l}(a\otimes b)+\tilde{l}(c\otimes d)$ and $\tilde{l}(\lambda(v\otimes w)):=\lambda\tilde{l}(v\otimes w)$. Then check that this is well-defined and $l=\tilde{l}\circ\pi$.

Uniqueness follows because the diagram has to commute, i.e. any different choice for $\tilde{l}$ would contradict $l=\tilde{l}\circ\pi$