Transformation between two sets of transformations I am currently looking at this transformation:
$$\mathcal{L}(V,W) \to \mathcal{L}(W^*, V^*), \qquad T\mapsto T^*$$
Where $T^*$ is defined as
$$T^*: W^* \to V^*, \qquad \phi \mapsto \phi T.$$
Here, $W^*$ and $V^*$ denote the dual spaces of $W$ and $V$ respectively.
I've gotten as far as understanding $T^*$ (I think). For $\mathbb{w} \in W^*$ and $\mathbb{v} \in V$,
$$(T^*(\mathbb{w}))(\mathbb{v}) = \mathbb{w}(T(\mathbb{v}))$$
I'm pretty sure this is right, but I'm not really sure what to do with the next part. I'm having trouble wrapping my mind around the idea of 
$$\mathcal{L}(V,W) \to \mathcal{L}(W^*, V^*), \qquad T\mapsto T^*$$
since it's a transformation between two sets of transformations, one of which is $\mathcal{L}(W^*, V^*)$, which is in itself a transformation between two sets of transformations. How can I go about understanding this? Is there an easy way to visualize / comprehend this? Any tips would be greatly appreciated.
Note: The end goal is to show that this transformation is linear.
 A: Your definition of $T^*$ is correct. You take an element $\phi$ of $W^*$ which is a linear map from $W$ to $\mathbb{K}$ and map it to $\phi \circ T$ where $T$ maps first $V$ to $W$ and $\phi$ then maps $W$ to $\mathbb{K}$ as before, so this defines a linear map from $V$ to $\mathbb{K}$. 
You correctly wrote this in the formula
$$ T^*(\phi)(v) = \phi(T(v))$$
But this also means that we can immediately check if the map $T\rightarrow T^*$ is linear. Consider the expression $T_1 + \lambda T_2$. This is mapped to $(T_1 + \lambda T_2)^*$ by definition. But now we also know what this does because of the above formula:
\begin{align}
(T_1 + \lambda T_2)^*(\phi)(v) &= \phi((T_1 + \lambda T_2)(v)) =\phi(T_1(v)+\lambda T_2(v)) = \phi(T_1(v)) + \lambda \phi (T_2(v)) \\
&=T_1^*(\phi)(v) + \lambda T_2^*(\phi)(v)
\end{align}
which exactly means that $(T_1 + \lambda T_2)^* = T_1^* + \lambda T_2^*$ which proves linearity.
To your first question: I don't know if it is helpful to try to visualize those rather cumbersome spaces of linear maps between spaces of linear maps between spaces... and so on. When trying to prove linearity it is best to just go through the definitions straightforwardly, at least this is my experience. I hope that helped. 
