250111 VO Selected topics in functional analysis: Riesz spaces (2013S)

Many of the vector spaces over the reals occurring in practice possess a "naturally" given order structure, for example, the well-known sequence spaces and the spaces of (continuous, bounded, integrable)n real-valued functions. Already in the introductory courses the notions of a vector spaces as well as that of a (partially) ordered set are introduced in an axiomatic way, however, the interplay between these two structures is not addressed explicitly in most courses on linear algebra or functional analysis.

As usual for "combined" structures, the axiomatic basis for ordered vector spaces consists of the axioms for vector spaces and those for order structures, plus the appropriate axioms of compatibility of the two structures at hand.

Now a Riesz space is an ordered vector space which - viewed as an ordered set - even is a lattice, i.e., for any two given elements there exists a supremum and an infimum. All the examples mentioned above are Riesz spaces with respect to their natural order structures.

In this lecture the foundations of the theory of Riesz spaces, together with the "appropriate" (=positive linear) operators acting on them will be presented, together with important and instructive examples. Towards the end of the course, topological and locally convex Riesz spaces will be envisaged: Here, in addition, a topology compatible with the linear and order structures will be incorporated as a third ingredient.

As prerequisites, linear algebra and real analysis essentially are sufficient. Some functional analysis will make a couple of things easier to comprehend, yet is not to be viewed as essential - except for the very last part of the course involving topological vector spaces which, however, will not be part of the exams.