Hello,

my question is about a proofstep in the proof of the second geometric version of the Hahn-Banach theorem. The theorem is:

Let X be a normed, real or complex vectorspace. Let A,B non-empty, convex subsets of X such that the intersection of A and B is empty. Let A be closed and B compact.

Then there exists an , , such that

for all .

The proof starts as follows:

is convex and closed (that C is closed is showed later in the proof and can be assumed for now) with . So there exists a such that . By the first geometric version of Hahn-Banach there exists an such that for all .

That means: for all , . [...]. qed.

I don't understand the red part of the proof. How can one conclude that?

(I know that , but I have no idea of how to get the "-"sign infront of the r.)

Thanks in advance for your help.

Regards,

engmaths