Let $X$ and $Y$ be [Banach spaces](/page/Banach%20Space) and $T \in \mathcal{L}(X, Y)$. Then:
1. $\ker T^* = (\operatorname{Im} T)^\perp$,
2. $\ker T = {}^\perp(\operatorname{Im} T^*)$,
3. $\overline{\operatorname{Im} T} = {}^\perp(\ker T^*)$,
4. $\overline{\operatorname{Im} T^*}^{w^*} = (\ker T)^\perp$.