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Analysis on metric spaces aims to develop concepts of differentiability beyond smooth and linear settings. It involves a more abstract geometric framework in which the primary structure is a distance (metric). This perspective leads to a deeper understanding of classical theory and to its further extensions.
In this lecture, I will introduce several ideas from this area and trace their connections to the basic notions of Calculus. We will begin with classical differentiability, proceed to weak derivatives, and conclude with metric differentiability. The central theme of the lecture is the Rademacher theorem on the differentiability of Lipschitz functions. We will review its proof via weak derivatives and discuss metric versions of this remarkable result.

More information: https://web.math.nccu.edu.tw/mfuchs/