I have some questions that do interest me, but professor does not want to discuss them due to them being hard. That is why I ask for reference here.
What is the dual of $L^{+\infty}(X, \mu)$, e.g. the space of all bounded functions in given measurable space X along with measure $\mu$?
What is the dual of $l^{\infty}$, e.g. the space of all bounded sequences of real numbers?(for those defining this as a sequence of complex numbers, if we have a sequence of complex numbers $z_1,...,z_n,...$, then for each $j \in \mathbb{N}$ we could write $z_j = a_j + ib_j$, where $a_j,b_j \in \mathbb{R}$ and watch a real sequence $a1,b_1,a_2,b_2,....,a_n,b_n$)
What are fundamental set in $L^{+\infty}$ and in $l^{+\infty}$?
What I know is the theorem stating that for any Banach space $X$, it could be embedded in $(X^*)^*$, so in first two questions it must hold $l^1 \subset (l^{+\infty})^*$ and as well $L^1 \subset (L^{+\infty})^*$, where the subset notation is meaning the images under those embedding I mentioned before. And about the last question I do not know anything smart, I only have the idea that it has to be extremely huge. Thanks for help!
