I'm trying to find out which Banach spaces $X$ satisfy the property: There exist $c>0$ and a continuous linear isomorphism $T:X\rightarrow X^*$ such as for any $x\in X, |<T(x),x>|\geq c \|| x \|| ^2$.
All Hilbert spaces satisfy that... I would like more 'exotic' examples, especially spaces that are not $p$-smooth.