Equivalent norms

Abstract

In this chapter we discuss the basic notion of an equivalent norm (Definition 2 above) in Section 3.1, and for finite-dimensional spaces, in Section 3.2.We consider then several basic constructions that provide equivalent norms. The first one works in the dual space, and generates the new equivalent norm on the original space (Section 3.3). The second one considers norming subspaces of the dual (Sections 3.4 and 3.5). The third one combines, in a way or another, some convex bodies in the space to produce an equivalent norm there (Section 3.6).