Locally r-Convex Tvs

To describe appropriately several phenomena which frequently appear when considering spaces derived from classical analysis, we are now going to introduce several subclasses of general tvs by taking into account some sort of convexity. More precisely, we will consider tvs admitting a 0-basis consisting of so-called r-convex sets, 0 < r ≤ 1. The most significant case occurs when r = 1, and it is in fact this case which supports a theory rich enough to handle properly the most important spaces originating from analysis. Actually, the remaining part of this book is essentially devoted to the study of such spaces which are known as locally convex spaces. Nevertheless, non-locally convex spaces having a 0-basis of r-convex sets for some 0 < r < 1 do occur; we shall in particular meet them when studying ideals of operators between Banach spaces in Chapter 19. For such reasons, and also since it does not require much extra work, we do not simply stick to the case r = 1.