Let $X$ be a normed vector space and $T$ a subset of $X^{\prime} = \mathcal{L}(X,\mathbb{R})$. Then define the set: $$^{\circ}T\ :=\ \{\;x\in X\ :\ F(x)=0,\ \forall\ F\in T\;\}.$$ (When) Is possible the get a bijection $G\in\mathcal{L}(T,\;^{\circ}T)$? Where $$\mathcal{L}(X,Y)\ =\ \{F:X\rightarrow Y\ \mbox{ such that } F \mbox{ is linear and bounded}\}.$$
Thanx so much in advance.