Set-theoretic methods in Banach spaces

With its broad applications in such sciences as physics, engineering, biology or medicine, analysis constitutes one of the main branches of modern mathematics. It is intensively studied by many mathematicians, however it happens often that some hypotheses of analysis cannot be proved as true or refuted as false with only use of analytical techniques. The problem usually lies in an assumed collection of axioms of mathematics i.e. sentences taken as true without a proof and from which every theorem of mathematics is derived concerning the notion of a set. Sets are considered as the most minimal objects in mathematics: all other objects such as numbers, functions or spaces may be built out of sets. However, despite the minimal character of sets, even a slightest change in the collection of the axioms may have a great impact on whole mathematics, implying that some mathematical objects will start or will stop to exist, some relations between objects will start or will stop to hold, or various object will start or will stop to have certain properties. The branch of mathematics studying properties of axioms of sets and their impact on the rest of mathematics is called set theory. In this project we are interested in investigating what is the impact of set theory on the existence of analytical spaces with certain properties. We especially ask from the set-theoretical point of view about the structure of spaces with various properties concerning convergence of infinite sequences of elements of infinite- dimensional spaces (so-called Banach spaces) to other elements, boundedness of infinite subsets of those spaces, or relations between the structure of spaces themselves and their elements. The knowledge about this structure allows to answer such questions as: how big a certain space is?, what other properties does it have?, and does it exist?. Methods of cardinal characteristics of the continuum will constitute our main research tool. These are set-theoretical techniques using objects that are a bit more complicated than sets such as certain families of infinite sequences of natural numbers or families of special infinite subsets of the set of real numbers. Those techniques have been deeply studied in set theory and now are well-understood. One of the main innovative aspects of the project is to use them in the investigation of the above-mentioned problems concerning Banach spaces. This application will reveal the combinatorial structure of certain Banach spaces used in analysis, and hence improve our understanding of them, but also will allow to prove that various important questions concerning them are undecidable, that is, depending on the assumed set of axioms they can be either proved or refuted.

With its broad applications in such sciences as physics, engineering, biology or medicine, analysis constitutes one of the main branches of modern mathematics. It is intensively studied by many mathematicians, however it happens often that some hypotheses of analysis cannot be proved as true or refuted as false with only use of analytical techniques. The problem usually lies in an assumed collection of axioms of mathematics i.e. sentences taken as true without a proof and from which every theorem of mathematics is derived concerning the notion of a set. Sets are considered as the most minimal objects in mathematics: all other objects such as numbers, functions or spaces may be built out of sets. However, despite the minimal character of sets, even a slightest change in the collection of the axioms may have a great impact on whole mathematics, implying that some mathematical objects will start or will stop to exist, some relations between objects will start or will stop to hold, or various object will start or will stop to have certain properties. The branch of mathematics studying properties of axioms of sets and their impact on the rest of mathematics is called set theory. In this project we were interested in investigating what is the impact of set theory on the existence of analytical spaces with certain properties. We especially asked from the set-theoretical point of view about the structure of spaces with various properties concerning convergence of infinite sequences of elements of infinite- dimensional spaces (so-called Banach spaces) to other elements, boundedness of infinite subsets of those spaces, or relations between the structure of spaces themselves and their elements. The knowledge about this structure allows to answer such questions as: how big a certain space is?, what other properties does it have?, and does it exist?. Methods of cardinal characteristics of the continuum constituted our main research tool. These are set- theoretical techniques using objects that are a bit more complicated than sets such as certain families of infinite sequences of natural numbers or families of special infinite subsets of the set of real numbers. Those techniques have been deeply studied in set theory and now are well-understood. One of the main innovative aspects of the project was to use them in the investigation of the above-mentioned problems concerning Banach spaces. This application reveals the combinatorial structure of certain Banach spaces used in analysis, and hence improves our understanding of them, but also allows to prove that various important questions concerning them are undecidable, that is, depending on the assumed set of axioms they can be either proved or refuted.