I m taking a course in functional analysis. The book state that the dual space of $l^1$, the set of real valued absolutely summable sequence, is $l^\infty$. Can anyone explain why the dual space of $l^1$ is $l^\infty$. I read a proof online http://math.uga.edu/~clayton/courses/608/608_5.pdf (Wayback Machine). I don't understand the correspondence between $l^1$ and $l^\infty$ they mentioned. Can some one explain more about this.