Application of Riesz representation theorem into dense subset

Question

Let $\mathcal{H}$ be a Hilbert space with an inner product $\langle\cdot,\cdot\rangle$ and $V\subset\mathcal{H}$ be a dense subspace. We already know that
$$\mathcal{H}^*=\{\langle v,\cdot\rangle|v\in\mathcal{H}\}$$
from the Riesz representation theorem where upper star is a continuos dual. Is there any way to represent the element $V^*$ as such form? More precisely, I want to know that if we define
$$\tilde{V}:=\{\langle v,\cdot\rangle|v\in V\},$$
,then $\tilde{V}$ could be a dense subset of $V^*$. Since $V$ might not be a Hilbert space, the Riesz-representation theorem doesn't work for $V$.

Answer 1

Any continuous linear functional on $V$ extends uniquely to a continuous linear functional on $H$ and it can be expressed as $v \to \langle v, x \rangle$ for some $x \in H$. Of course, $x$ need not be in $V$.