I don't understand in the answer of 11b, why does that result follows from 11a?
I know that the orthogonal complement is always a closed set, and that for closed sets, the orthogonal complement of the orthogonal compelement - is the original set. And yet, how can I derive the given result from it?
In (11.a) replace $L_1$ by $L_1 ^\perp$ and $L_2$ by $L_2 ^\perp$. You will get $(L_1 ^\perp + L_2 ^\perp) ^\perp = (L_1 ^\perp) ^\perp \cap (L_2 ^\perp) ^\perp$. Remeber now that, in general, $(L^\perp)^\perp = \overline L$. Since in (11.b) $L_1$ and $L_2$ are assumed closed, the previous equality becomes $(L_1 ^\perp + L_2 ^\perp) ^\perp = L_1 \cap L_2$. Taking the orthogonal complement, this becomes $[(L_1 ^\perp + L_2 ^\perp) ^\perp] ^\perp = (L_1 \cap L_2) ^\perp$. Since the left-hand side is $\overline {L_1 ^\perp + L_2 ^\perp}$, you get your result.