About adjoint of densely defined linear operator

Let's consider $\mathfrak{H}$ be a Hilbert space, a densely defined linear operator on $\mathfrak{H}$ is an ordered pair $(T,D_T)$, where $D_T$ is a dense linear subspace of $\mathfrak{h}$, and $T:D_T\rightarrow\mathfrak{h}$ is a linear transformation.

In the text, it says using Rieze representation theorem, for $g\in D_T*$ , there exists ''unique'' element $T^*g\in\mathfrak{h}$ such that $\langle Tf,g\rangle=\langle f,T^*g\rangle$.

I don't know why it should be unique.

Thank for your help!