This chapter assembles the necessary concepts and statements on metric spaces and Banach spaces which will be used throughout the book. Separability of these spaces will play an important role for their usage in probability theory. It turns out to be useful to consider not only Banach spaces but also quasi-Banach spaces and p-normed Banach spaces. Linear operators between Banach spaces are introduced and the equivalence of continuity and boundedness is shown. The dual space of a Banach space is defined, and for the convenience of the reader, the Hahn-Banach theorem, a separation theorem and the closed graph theorem are listed.