Is every Banach space densely embedded in a Hilbert space? Can every Banach space be densely embedded in a Hilbert space? This is clear if the Banach space is actually a Hilbert space, but much can you relax this?
If the embedding exists, is the target Hilbert space unique?

Suppose that there exists an injective, bounded linear operator $T\colon X\to H$, where $H$ is a Hilbert space. Then $\|x\|_{\rm new}=\|x\|+\|Tx\|$ ($x\in X$) defines an equivalent, strictly convex norm on $X$.
Not every Banach spaces admits a strictly convex renorming -- just to mention $\ell_\infty(\omega_1)$ (Day) or $\ell_\infty / c_0$ (Bourgain).
