I have been studying tensor calculus by myself, but I have found the following claim in my book:
The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is canonically isomorphic to the space $\mathcal{L}_{p}(V)$ of $p$-linear forms over $V$, i.e. this isomorphism maps the tensor product $f^{1} \otimes \cdots \otimes f^{p} \in V^{0}_{p}$ of linear forms $f^{i} \in V^{*}$ to the usual product $f^{1} \cdot f^{2} \cdots f^{p} \in \mathcal{L}_{p}(V)$, which is defined by $(f^{1} \cdot f^{2} \cdots f^{p})(v_{1},v_{2},\ldots,v_{p}) = f^{1}(v_{1}) \cdot f^{2}(v_{2}) \cdots f^{p}(v_{p})$. Schematically: $$\mathcal{L}_{p}(V) \to V^{*} \otimes \cdots \otimes V^{*}$$ $$f^{1} \cdot f^{2} \cdots f^{p} \to f^{1} \otimes f^{2} \otimes \cdots \otimes f^{p}$$
I tried to understand this with an example, but I wasn't able to get it. I would appreciate any help!