Banach spaces of continuous and Lipschitz functions

The main purpose of our project is to study spaces of so-called continuous and Lipschitz functionsspecial kind of mathematical spaces consisting of regular and nice functionsin the context of different mathematical fields like geometry, topology, analysis and logic. Intuitively speaking, continuous functions are those functions between two given mathematical spaces whose graphs do not have holes or sudden jumps. Lipschitz functions are a particular type of continuous functions, where the degree of the slope of the function is everywhere bounded by a fixed positive number, so the graph of the function seems to be regular and tame. The spaces on which we consider these continuous and Lipschitz functions are so-called metric spaces. The main motivation for studying these metric spaces is to to generalize the physical space in which we live to more general settings. Here both the structure of the space and the way of measuring distances can be different from our everyday experience. Thus, our project can be thought as devoted to study properties of transformations (functions) between two special generalizations of our physical world. In the recent decades a geometrical theory of these spaces, called metric geometry, has been developed and appeared to be very fruitful and promising. The continuous and Lipschitz functions we consider form so-called Banach spacesa special type of vector spaces where in addition to adding functions and multiplying them with numbers, we are able to introduce distances between functions and measure them. Introducing different types of distances between functions causes that those Banach spaces have also different geometrical properties. Thus, in relation to our aim of studying properties of metric spaces on which functions live we also plan to study properties of the Banach spaces built over these functions. As we already mentioned, we plan to use techniques and methods from different areas of mathematics such as geometry, analysis or logic. Such an interdisciplinary approach seems to be completely innovative and thus we believe that our project will allow us to deepen our understanding of those Banach spaces and thus the metric spaces over which they are built.

The main purpose of our project was to study spaces of so-called continuous and Lipschitz functionsspecial kind of mathematical spaces consisting of "regular" and "nice" functions between underlying spaces having some special shape (so-called "topology") or distance function (so-called "metric")in the context of different mathematical fields like geometry, topology, analysis, or logic. In addition to studying those spaces of functions, we have been addressing the question of how much geometry or topology can be found and understood in the underlying spaces and which geometric and topological properties these spaces have. For some of the spaces of functions studied by us, we discovered how the local properties of the underlying geometric spaces together with a very rough picture of the global structure come together to describe the spaces of functions in more detail. In addition, we showed that some of these function spaces cannot be transformed onto some of the others by continuous transformations as well as we established how some special infinite sets in these spaces look like and behave. Using techniques and methods from different areas of mathematics, we were not only able to obtain a deeper understanding of these function spaces and the underlying metric or topological spaces but also discovered something new about the methods we used.