Now I do understand that the question might involve using Riesz representation as it involves a linear functional and we know that it can be written using an inner product. So $u$ is the representer of the linear functional in this case. So I think I might be able to find $u$ iff I know the orthonormal basis of $V$(which is not given). That brings me to my current roadblock.
Also even if I were to assume the orthonormal basis how would I use it to figure out $T^*$($\alpha$). Is there some kind of relation between them.
PS: I know what adjoint is but how do I use it to figure out the adjoint of $T^*(\alpha)$.
By definition $\langle T^{*}\alpha, x \rangle= \alpha Tx$ since the inner product of two scalars is just their ordinary product. Hence $\langle T^{*}\alpha, x \rangle= \alpha \langle x, u \rangle$ which shows that $T^{*}\alpha=\alpha u$. [This proof is when the scalar field is $\mathbb R$. I will let you show that the answer is same when the scalar field is $\mathbb C$].