Linear maps using Tensor Product While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation:
"To give a linear map $V \rightarrow V$ is the same as to give a linear map $V^* \otimes V\rightarrow \mathbb{R}$, assuming we're looking at real vector spaces."
Could anybody kindly explain the above sentense in detail with an example?   I am not a math-major, but very much interested in tensor analysis.  Thank you in advance. 

Not so detailed as you want but I'll give a hint. 
It means you have a vector space isomorphism $\mathcal{L}(V^*\otimes V, \mathbb R)\simeq \mathcal{L}(V)$. For seeing this you have to define a linear bijective map $$\Phi:\mathcal{L}(V)\longrightarrow \mathcal{L}(V^*\otimes V, \mathbb R).$$ To each $T\in \mathcal{L}(V)$ you assign $\Phi_T:V^*\otimes V\longrightarrow \mathbb R$ given by: $$\Phi_T(f\otimes v)=f(Tv).$$ So far you have defined $\Phi_T$ only for pure tensors (those of the form $f\otimes v$, $f\in V^*$ and $v\in V$) so you must define to all $V^*\otimes V$. Since you want $\Phi_T$ to be linear you simply extend it by linearity. 
It is an exercise showing $\Phi$ is linear and bijective.
