Let $V$ be a finite-dimensional vector space over a field $F$, and $B:V\times V\to F$ be a bilinear map. Let a linear map $α_B:V→V^∗$ to the dual space of $V$ be given by $v \mapsto [w \mapsto B(v,w)]$.
Calculate the composition $V\overset{ιV}{\mapsto}V^{**}\overset{α^T_B}{→}V^∗$, where $ιV:V\overset{∼}{\mapsto}V^{**}$ is the canonical isomorphism.
Honestly, this kind of problem seems very tough for me so just have fun with it and I hope anybody can help me, at least give me a clue how to start..
For $v \in V$ the image of $v$ under the canonical isomorphism is the linear form of defined on $V^*$ by
$$\begin{array}{l|rcl} (V^*)^* \ni \iota(v) : & V^* & \longrightarrow & F \\ & \phi & \longmapsto & \iota(v)(\phi) = \phi(v)&\end{array}$$
And by definition of the transpose, $\alpha_B$ is the application
$$\begin{array}{l|rcl} \alpha_B^T : & (V^*)^* & \longrightarrow & V^* \\ & \psi & \longmapsto & \alpha_B^T(\psi)&\end{array}$$ defined by
$$\alpha_B^T(\psi)(v) = \psi(\alpha_B(v))$$ for all $v \in V$. We have to identify $ \alpha_B^T \circ \iota$ which is an element of $\mathcal L(V, V^*)$, i.e. for $v \in V$, $(\alpha_B^T \circ \iota)(v) $ is a linear form of $V$ which maps an element $w \in V$ to $[(\alpha_B^T \circ \iota)(v)](w) \in F$. More precisely:
$$[(\alpha_B^T \circ \iota)(v)](w) = \iota(v)(r \mapsto B(w,r))=B(w,v).$$
Finally
$$(\alpha_B^T \circ \iota)(v) = [w \mapsto B(w,v)]$$ or using the terminology of the question, $\alpha_B^T \circ \iota$ is given by $v \mapsto [w \mapsto B(w,v)]$.