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?



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


