What is the $\epsilon^{00}$-topology?

"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.

Solution 1:

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