Let $E$ be a $n$-dimensional vector space. Prove that: $$\begin{array}{ccll} \Phi:&Hom(E\times \stackrel{(r)}{\ldots} \times E,E)&\longrightarrow&\bigotimes_r^1 E\\ &\psi &\longmapsto &\Phi(\psi) \end{array}$$ given by $\big(\Phi(\psi)\big)(u_1,\ldots,u_r,\omega)=\omega\big(\psi(u_1,\ldots,u_r)\big)$ for all $u_1,\ldots,u_r\in E$ and $\omega \in E^*$ is an isomorphism.
(1) In fact, the linearity of $\psi$ and $\omega$ prove that $\Phi$ is linear.
(2) Also, it is easy to show that is injective. Let $\psi_1,\psi_2\in Hom(E\times \stackrel{(r)}{\ldots}\times E,E)$ such that $\Phi(\psi_1)=\Phi(\psi_2)$. Then for all $u_1,\ldots u_r\in E$: $$\big(\Phi(\psi_1)\big)(u_1,\ldots,u_r,\text{id}_E)=\big(\Phi(\psi_2)\big)(u_1,\ldots,u_r,\text{id}_E)\Rightarrow \psi_1(u_1,\ldots,u_r)=\psi_2(u_1,\ldots,u_r)$$ And $\psi_1$ must be equal to $\psi_2$
(3) How can I get that $\Phi$ is surjective??
First note that this wouldn't be an isomorphism if $E$ was infinite-dimensional. So we have to use the fact that $E$ is finite-dimensional somewhere in the proof. The way to do this is just to verify that both sides have the same dimension $\left((\mathrm{dim}E)^{r+1}\right)$ and hence an injection between them is an isomorphism.