Harry Poppe

An abstract Alaoglu theorem

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 68, 65 - 69 (2013)

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}).

Keywords:   Duality theory, Compactness in Banach (or normed) spaces

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