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RESEARCH PRODUCT

The Theory of Normed Modules

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This chapter is devoted to the study of the so-called normed modules over metric measure spaces. These represent a tool that has been introduced by Gigli in order to build up a differential structure on nonsmooth spaces. In a few words, an \(L^2({{\mathfrak {m}}})\)-normed \(L^\infty ({{\mathfrak {m}}})\)-module is a generalisation of the concept of ‘space of 2-integrable sections of some measurable bundle’; it is an algebraic module over the commutative ring \(L^\infty ({{\mathfrak {m}}})\) that is additionally endowed with a pointwise norm operator. This notion, its basic properties and some of its technical variants constitute the topics of Sect. 3.1.