The completeness of locally convex space generated by relatively weakly $p$-compact sets

Question

Let $X$ be a Banach space and $1\leq p<\infty$. A bounded subset $K$ of $X$ is relatively weakly $p$-compact if $K$ is contained in $S(B_{l_{p^{*}}})$for some operator $S$ from $l_{p^{*}}$ into $X$. Let $\mathcal{F}$ be the family of all relatively weakly $p$-compact subsets of $X$. For $K\in \mathcal{F}$, define a semi-norm $p_{K}$ on $X^{*}$ by $p_{K}(x^{*})=\sup_{x\in K}|<x^{*},x>|,x^{*}\in X^{*}$. The locally convex topology generated by $\{p_{K}:K\in \mathcal{F}\}$ is denoted by $\rho^{*}_{p}$. My question: Is the locally convex space $(X^{*},\rho^{*}_{p})$ complete?

Answer 1

I think that one can show that the space $(X^*, \rho_p^*)$ is complete on the following lines:

(1) Consider an arbitrary Cauchy net $x^*_\alpha\in X^*$ in the described topology. This assumption implies that the net is pointwise convergent (I mean that $x_\alpha^*(x)$ are convergent nets of scalars for all $x\in X$.) Thus the net has a limit $x^\sharp$ which is a linear (not necessarily continuous) functional. If it belongs to $X^*$ for each such net, we are done.

(2) Suppose that for one of the nets the limit is not in $X^*$. Then it is an unbounded linear functional and thus there is a sequence of vectors $\{x_n\}_{n=1}^\infty$ in the unit ball of $X$ such that $|x^\sharp(x_n)|\ge 4^n$. Then the sequence $\{2^{-n}x_n\}$ is rapidly convergent to $0$ and thus is covered by an image $K$ of an operator from $\ell_{p^*}$. The corresponding seminorm is infinite on $x^\sharp$ and thus elements of $X^*$ cannot converge to it in the considered topology.