Dual cone and sum of closed cones

Question

Picture below is from the 35 page of Schneider R.-Convex Bodies_ The Brunn-Minkowski Theory-Cambridge University Press (2013) , I think $C^o$ is always closed no matter $C$ is closed or not. Because the inner product is continuous. Besides the I think the sum of closed convex cones must be closed, because the sum is continuous . Where is my mistake ?

Answer 1

Yes, $C^\circ$ is always closed.

However, the sum of the closed convex cones $$A = (-\infty,0] \times \{(0,0)\} \\ B = \operatorname{cl-cone}(B_{1}(1,1,0))$$ is not closed. Here, $B_{1}(1,1,0))$ is the closed ball centered at $(1,1,0)$ with radius $1$, touching all three coordinate planes.

Answer 2

I just think the answer of gerw is a little complex. So I according his way make the below picture . Green and Black are two closed cones , but the sum of them is open ( the sum is $R^2$).