I am currently reading Jesse Peterson's lecture notes on von Neumann algebras. I'm confused by lemma 4.4.2. In particular, it seems to me that Hahn Banach theorem here can only conclude the map $\phi$ is in the X** not in X. Without this it seems the subsequent proof of uniqueness of predual of von Neumann algebra cannot go through. Can somebody help? Thank you very much.
The proof is using the version of the Hahn-Banach theorem which works for locally convex spaces. We have a $C^*$-algebra $A$ and a Banach space $X$ with $X^*=A$. We then equip $A$ with the weak$^*$-topology, that is, $\sigma(A,X)$. As the proof notes, $(A)_1 \cap A_+$ is $\sigma(A,X)$ compact, so if $a<0$ we can find a $\sigma(A,X)$-continuous linear functional which separates. But $\sigma(A,X)$-continuous linear functionals are simply elements of $X$.
I agree that this is a little unclear in the proof. What's a good book for the Hahn-Banach theorem at this level of generality? You could look at Rudin, Functional Analysis. Or I really like Conway's Functional Analysis. I think Theorem 3.9 of Chapter IV in the 2nd edition is enough. I'm afraid I don't know a good online resource (can anyone suggest one?)