Stereotype locally convex spaces (Q2710694)

The author gives complete proofs of some results on stereotype locally concise spaces which where announced in [\textit{S. S. Akbarov}, Mat. Zametki 57, No. 3, 463-466 (1995); Engl. transl. Math. Notes 57, No. 3, 319-322 (1995; Zbl 0836.46002)].NEWLINENEWLINENEWLINEA LCS space \(X\) (locally convex space) is called stereotype if it is pseudocomplete and pseudosaturated, this is:NEWLINENEWLINENEWLINE(i) every totally bounded Cauchy net converges in \(X\);NEWLINENEWLINENEWLINE(ii) every closed balanced set \(D\subset X\) with the property that for every totally bounded subset \(S\subset X\) there is a neighbourhood \(U\) of zero in \(\mathbb{X}\) such that \(S\cap U\subseteq M\), is itself a neighborhood of zero.