I'm currently reading Nakahara's Geometry, topology and physics (about tensors), and came across with the following proposition (exercise 2.12, p.99):
Show that a linear map $f:V \to W$ is a (1,1) tensor.
As requested, a $(n,m)$-tensor is defined in the book as a multilinear map \begin{equation} T: \otimes_{n} V^{*} \otimes_{m} V \to \mathbb{R} \end{equation} where $V$ is a vector space, $V^{*}$ it's dual space. For example a mapping $T_{1,2} = V^{*} \times V \times V \to \mathbb{R}$ is a $(1,2)$-tensor. At least I found the notation confusing, as to how $\otimes_{1} V$ and the like should be interpreted, hence the example. Any help with understanding the notation would be appreciated as well. It's also difficult to visualise what it means to form have an object with both vectors and linear maps as parts of its components.
I'm not sure even how to begin. We'd want to show that $f:V\to W \Rightarrow f: V'^{*}\times V' \to \mathbb{R}$? I also know that, since $\mathrm{dim}V' = \mathrm{dim}V'^{*}$, the spaces are isomorphic and I can further write the latter as $f:V'^{2}\to \mathbb{R}$, and $V'^{2}$ is also a vector space, say, $V''$. Then if we mark $V''=V$, and think about the project mapping (? not sure about the terminology) $f_i:V\to \mathbb{R}$ such that $f_i(v)=w_i$, where $v\in V$ and $w\in W$. Then that mapping is specified by $f$, but it's still not $f$ that would be the tensor.
Edit: added that $f$ in the proposition is a linear map.