| MSC: | 46A20 | Duality theory |
| 46B50 | Compactness in Banach (or normed) spaces |
Abstract:
We start with the paper \cite{pop6} and repeat the definition of the dual space
$X^d$ of the space $X$ w.\,r.\,t.\,another space $Y$. By Corollary~3.3 of this paper
was given a (strong) generalization of the Alaoglu theorem for normed spaces.
The Alaoglu theorem concerns the compactness of subsets $H\subseteq X^d$, where for $H$ the pointwise
topology $\tau_p$ is considered. Of course for the classical Alaoglu theorem we have:
\begin{gather*}
(X,\|\cdot\|) \text{ is a Banachspace},\quad
Y=\mathbb{K},\quad X^d=X',\quad H=B(X')\subseteq X',
\end{gather*}
the norm-closed ball, and the topology $\tau_p$ here is nothing else than the weak-star topology.
What is the aim of this paper? We prove a general $\tau_p$-compactness theorem
for special subsets $H\subseteq X^d$ (Theorem 2.3). This theorem includes both the generalized
Alaoglu theorem of \cite{pop6}, Corollary~3.3 and the Alaoglu theorem
for locally convex topological vector spaces (see for instance \cite{pop5}).