Metric Spaces

As calculus developed, eventually turning into analysis, concepts first explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. It is a set on which a notion of distance between each pair of elements is defined, and in which notions from calculus in R (open and closed intervals, convergent sequences, continuous functions) can be studied. Many of the fundamental types of spaces used in analysis are metric spaces (e.g., Hilbert spaces and Banach spaces), so metric spaces are one of the first abstractions that has to be mastered in order to learn analysis.