Proposition 16.2.3.label Let $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$, then:
- (1)
$\emptyset^{\circ} = F$ and $E^{\circ} = \bracs{0}$.
- (2)
For any $\alpha \in K \setminus \bracs{0}$ and $A \subset E$, $(\alpha A)^{\circ} = \alpha^{-1}\cdot A^{\circ}$.
- (3)
For any $\seqi{A}\subset E$, $\paren{\bigcup_{i \in I}A_i}^{\circ} = \bigcap_{i \in I}A_{i}^{\circ}$.
- (4)
For any $A \subset B \subset E$, $A^{\circ} \supset B^{\circ}$.
- (5)
For any saturated ideal $\sigma \subset \mathfrak{B}(E, \sigma(E, F))$, $\bracs{S^\circ|S \in \sigma}$ is a fundamental system of neighbourhoods at $0$ for the $\sigma$-uniform topology on $F$.
Proof. (2): For any $\lambda \in K \setminus \bracs{0}$ and $A \subset E$,
(5): Let $S \in \sigma$, then
Since $\sigma$ is saturated, $\bracs{S^\circ|S \in \sigma}$ is a fundamental system of neighbourhoods at $0$ for the $\sigma$-uniform topology.$\square$