Let $E$ be a [[Separable Topological Space|separable]] $\real$-[[Banach Space|Banach space]], $H \subset E$ be a $\real$-[[Hilbert Space|Hilbert space]] [[Bounded Linear Map|continuously]] embedded as a [[Dense|dense]] subspace of $E$. 1. For each $\phi \in E^*$, there exists a unique $y_\phi \in H$ such that $\phi|_H = \angles{\cdot, y_\phi}_H$. The mapping $E^* \to H$ defined by $\phi \mapsto y_\phi$ is an injective [[Continuous Linear Map|continuous linear map]] with respect to the [[Weak Topology|weak* topology]] on $E^*$ and the weak topology on $H$, whose image is dense in $H$. 2. If $x \in E$, then $x \in H$ if and only if