Barrelledness and Reflexivity

Barrels have been introduced in 8.3 as polars of weakly bounded sets, and we know already that every closed, absolutely convex O-neighbourhood is a barrel. If the converse is also true, then the lcs under consideration is said to be barrelled. This is the case e.g. for Fréchet spaces and many other important lcs. Among the most important results on barrelled lcs to be proved in 11.1 we mention the Banach-Steinhaus theorem and Pták’s extension of the classical open mapping theorem. 11.2 is devoted to the larger class of quasi-barrelled lcs. In 11.3 we discuss the permanence properties of these spaces, with special emphasis on the problem of subspaces. Semi-reflexive and reflexive lcs are investigated in 11.4, and 11.5 is devoted to the study of semi-Montel and Montel spaces. 11.6 contains some of the most important facts on Fréchet-Montel spaces. In 11.7 we consider these concepts again for spaces of continuous functions and determine in particular the barrelledness character of C(X) for both, the compact-open and the pointwise topology, X being a completely regular space. Mainly for the purpose of appropriate calculation of the duals of spaces of integrable functions, we have included in 11.8 some elementary details on uniformly convex spaces. The chapter concludes with some fundamental facts on Hilbert-spaces.