At the moment I am preparing my doctoral thesis and writing down results I have already obtained so that I can use them later. In the process, I have found myself using both the notation $X^*$ and $X'$ for the continuous dual space of normalized spaces, depending on which literature I am currently working with. I realize that I should settle on one notation. But which is the better one?
According to Wikipedia and most of the literature I use, $X^*$ denotes the algebraic dual space and $X'$ denotes the continuous dual space. However, I also find many books where the continuous dual space is denoted by $X^*$.
I tend (for aesthetic reasons) to use the notation $X'$. But then wouldn't it be appropriate to write weak' convergence and $x_n'\rightharpoonup'x'$ instead of weak* convergence and $x_n^*\rightharpoonup^*x^*$? However, this seems to me to be extremely unusual. Is there a standard work that I can use as a guide?