I'm trying to solve the problem 5.12 of Harim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations; but I'm stucked understanding the statement which comes as follows:
Let $E$ be a vector space equipped with the scalar product $( , )$. One does not assume that $E$ is complete for the norm $|u| = (u, u)^{1/2}$ ($E$ is said to be a pre-Hilbert space).
Recall that the dual space $E^*$, equipped with the dual norm $||f||_E^*$, is complete. Let $T : E \rightarrow E^*$ be the map defined by $ \langle Tu, v\rangle_{E^*,E} = (u, v) \hspace{5mm} \forall u, v \in E. $ Check that $T$ is a linear isometry. Is $T$ surjective? Our purpose is to show that $R(T)$ is dense in $E^*$ and that $||\hspace{2mm}||_{E^*}$ is a Hilbert norm.
I don't quite understand how does $T$ work. Does it take as arguments both $u$ and $v$ or just $v$? Is it using the Riesz-Fréchet representation theorem implicitly? Is $T \in E^*$?