In my studies of convexity I have recently come across the following:
Let $ V=\mathbb{R}^d ; d \geq 1 $ be a Euclidean space and $ S \neq \emptyset $ be a non empty set of the vector space $ V $, where we define its dual as usual $ S^* = \{y\in V | \forall x \in S ; \langle y,x \rangle \geq 0 \} $. Now I have already proved that $ S^* $ is always a closed convex cone, now I wish to prove the following: $ S^* = (\bar{S})^* = {(conv(S))}^*= {(cone(S))}^* $ where "conv" is the usual convex hull and "cone" is the minimal convex cone containing $S$ and $ \bar{S} $ is the closure of $S$.
I have proved that the dual set is always a closed convex cone, is there a way to prove these equalities I have written, I seem to be stuck on this. All help is appreciated.
Assume that $S$ has nonzero vector
Note that for $x\neq 0\in S$ $H_x:=\{ y|y\cdot x\geq 0\}$ is closed half space That is $ S^\ast$ is intersection of these closed half space Hence $S^\ast$ is closed convex In further closed half space is cone so that $S^\ast$ is cone
(1) $S^\ast = (\overline{S})^\ast$ : From this argument we have that $$ S\subset \overline{S}\Rightarrow S^\ast \supseteq (\overline{S})^\ast $$
If $x_n\in S$ and $x_n\rightarrow x$ let $$ v\in \bigcap_{n} H_{x_n} $$
so that $$ v\cdot x_n\geq 0 \Rightarrow v\cdot x\geq 0 $$
That is $v\in H_x$ so that $\bigcap_{n} H_{x_n} \subset H_x $
(2) $S^\ast =({\rm conv}\ S)^\ast$ : $x_i\in S$ then $v:=\sum_{i=1}^n c_ix_i\in {\rm conv}\ S$ where $c_i\geq 0,\ \sum_i c_i=1$ Assume that $$w\in \bigcap_i H_{x_i}$$ Then $$ w\cdot v =\sum_i c_ix_i\cdot w \geq 0\Rightarrow w\in H_v$$
(3) $({\rm conv}\ S)^\ast =({\rm cone}\ S)^\ast$ : For $v\in {\rm cone}\ S$, we have $$ w\in {\rm conv}\ S,\ tw=v,\ t>0 $$
Hence $$ H_v=H_w$$