Suppose $H$ is a Hilbert space with scalar product $(\cdot , \cdot )$ and norm $| \cdot |$. Let $V \subseteq H$ be a dense linear subspace and say it has its own norm $\| \cdot \|$ such that $V$ is again Banach for $\| \cdot \|$.
And suppose the injection $V \rightarrow H$ is continuous, i.e, $|v| \leq C \|v\| \forall v \in V$. Consider the operator $T: H \rightarrow V^*$ such that:
$\langle Tu, v\rangle_{V^*, V} = (u, v)_H$ $\forall u \in H, \forall v \in V$. Now:
Prove $\|Tu\|_{V^*} \leq C|u|$ for all $u \in H$
T is injective
$R(T)$ is dense in $V^*$ if $V$ is reflexive
$f \in R(T) \iff \exists a\ge0$ such that $|\langle f, v\rangle_{V^*, V}|\leq a|v|$ for all $v \in V$
I feel stupid but I'm not immediately sure here of the rigor here even though intuitively I sorta see it:
Basically $T$ is $i^*$ where $i : V \rightarrow H$ is the inclusion. At this point I think you'll want to use the density of $V$ in $H$ to show injectivity of $T$ I think. But I'm not sure how to show $1$.
Then I'm not sure how to show 3. either.
For 4., $\implies$ is clear enough but not $\impliedby$. I think the idea is to define a new functional $\varphi : V \rightarrow \mathbb{R}$ as $\varphi(v) = \langle f, v\rangle$ and using the density of $V$ in $H$ to extend to all of $H$.
I would appreciate some help here, for filling in the blanks more formally.