TITLE: What is the $\epsilon^{00}$-topology?
QUESTION [11 upvotes]: "If $F$ is a locally convex space, then the topology on $F^*$ (the topological dual of F) will be the strong topology, and the topology on $F^{**}$ will be the $\epsilon^{00}$-topology." This is the first time I've seen this notation. It is described as the uniform convergence topology on the polars of p-unit balls, but I would like to know what is the exact thing this $\epsilon^{00}$-topology is referring to, it is also mentioned in some other papers without any further explanation - is it always the same thing explained here? 
The paper I'm reading right now was published in 1974, "Linear Operators and Vector Measures" by Brooks and Lewis.

REPLY [4 votes]: Before  stating  the  main  theorems  we  recall  some  facts  about  representing  measures  and  other topics.  $Y'$  is  the continuous dual  of $Y$,  $Y''$  the continuous  bi-dual.  Let  $σ(Y'', Y')$  denote  the $Y'$  topology  of $Y''$ and  let $\{q'\}$ be a family  of  seminorms generating this topology.  Since representing measures  can be  taken  as  being  defined  on $B(H)$ (see  [10]), $H$  a compact Hausdorff   space,  and  are  $L_c''(X,  Y)$-valued,  $xμ(\cdot)$  being  countably  additive  in  the $(Y''Y')$  topology  for  each  $x \in X$,  we  may  define  our  integral with  convergence in  $\{q'\}$   rather than $\{q''\}$,  where  $\{q''\}$   denotes the  family of  continous  seminorms  generating  the  topology  on  $Y''$  sometimes  called the  "$\epsilon^{00}$  topology"  [20,  p.  71],  which  is  the  usual  norm  topology  of   $Y''$ when  $Y$  is  a  normed  space.

https://msp.org/pjm/1984/111-1/pjm-v111-n1-p17-s.pdf