A locally convex space (lcs) is barrelled if every “barrel” (= closed, absolutely convex and absorbing subset) is a neighbourhood of the origin. The importance of such spaces is due to the fact that they admit extensions of the classical Banach-Steinhaus theorem, closed graph theorem…. Every Baire lcs is barrelled, but not conversely.
The present book deals with classes of lcs which are intermediate between Baire and barrelled lcs, for example Baire-like spaces, supra-barrelled spaces, totally barrelled spaces, classes of lcs which are labeled “barrelled of class \(n\)” (for \(n\in \mathbb{N}\) or \(n= {\mathcal X}_0\)), etc.
A major portion of the book is concerned with barrelledness properties of specific spaces. Among these is the (sup-normed) space \(\ell^\infty_0(\Sigma)\) of all \(\Sigma\)-simple functions on a set \(\Omega\), \(\Sigma\) being a \(\sigma\)-algebra (or just an algebra) of subsets of \(\Omega\), the vector valued analogue \(\ell^\infty_0(\Sigma, E)\), \(E\) an lcs, the Bochner spaces \(L^p(\mu, X)\), \(X\) a normed space, as well as spaces of strongly measurable and Pettis-integrable \(X\)-valued functions.