Let $V$ and $H$ be two Hilbert spaces such that $V \subset H$ with continuous and dense injection. Then we have $$V \subset H \cong H' \hookrightarrow V',$$ where $\cong$ means that there exists an isometric isomorphism and $\hookrightarrow$ means that there exists a continuous injective map.
My question is: can we always find a Hilbert $V^* \cong V'$ such that $H \subset V^*$ with continuous and dense injection ? This is for instance the case for $$ V=\left\{u \in {\mathbb{R}}^{\mathbb{N}}; \quad \sum_{n=1}^{+\infty} n^2 u_n^2<+\infty\right\}, \qquad H=\left\{u \in {\mathbb{R}}^{\mathbb{N}}; \quad \sum_{n=1}^{+\infty} u_n^2<+\infty\right\}, \qquad V^*=\left\{u \in {\mathbb{R}}^{\mathbb{N}}; \quad \sum_{n=1}^{+\infty} \frac{1}{n^2} u_n^2<+\infty\right\} .$$ I'm asking for a "real" inclusion, i.e. in the algebraic sense.