Lemma 9.1.3. Let $E$ be a TVS over $K \in \RC$, $A, B \subset E$ be convex, then the following sets are convex:
$A^{o}$.
$\ol{A}$.
$A + B$.
For any $\lambda \in K$, $\lambda A$.
Proof. (1): By Lemma 9.1.2.
(2): Let $x, y \in \ol{A}$. By Definition 4.5.2, 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 $tx + (1 - t)y \in \ol{A}$.$\square$