Bulletin de la Société Royale des Sciences de Liège (Bull. Soc. Roy. Sci. Liège) (19840101), 53, no.~1, 57-67. ISSN: 0037-9565 (print).eISSN: 1783-5720.
Subject
46 Functional analysis -- 46A Topological linear spaces and related structures 46A50 Compactness in topological linear spaces; angelic spaces, etc.
The author calls a subset $A$ of a topological vector space $E$ asymptotically compact if $([0,1]A)\cap V$ is relatively compact for some neighbourhood $V$ of zero. The asymptotic cone is $A\sb \infty=\break \bigcap\sb {\varepsilon>0}[0,\varepsilon]A$. The key fact is that $A$ is relatively compact if and only if $A$ is asymptotically compact and $A\sb \infty=\text{cl}\{0\}$. A calculus of asymptotic compact sets is developed. A variety of useful closure results due to J. A. Dieudonne [Math. Ann. 163 (1966), 1--3; MR0194865 (33 \#3071)] and J.-P. Dedieu [Bull. Soc. Math. France Mem. No. 60 (1979), 31--44; MR0562254 (83i:46052)] are considerably extended. An application is also given to tangent cone theory.