Lemma 11.1.5.label Let $E$ be a TVS over $K \in \RC$, $A, B \subset E$ be convex, then the following sets are convex:
- (1)
$A^{o}$.
- (2)
$\ol{A}$.
- (3)
$A + B$.
- (4)
For any $\lambda \in K$, $\lambda A$.
Proof. (1): By Lemma 11.1.4.
(2): Let $x, y \in \ol{A}$. By Definition 5.5.2, there exists filters $\fF, \mathfrak{G}\subset 2^{A}$ such that $\fF$ converges to $x$ and $\mathfrak{G}$ converges to $y$. In which case,
\[\fU = \bracs{tE + (1 - t)F|t \in [0, 1], E \in \fF, F \in \mathfrak{G}}\subset 2^{A}\]
converges to $tx + (1 - t)y$ by (TVS1) and (TVS2). Hence $tx + (1 - t)y \in \ol{A}$.$\square$