Partners: Functional Analysis and Topology

Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space—the space of continuous linear functionals—of a normed space played an especially important role in the formative years of functional analysis. To further this endeavor, many new kinds (weak, strong, etc.) of convergence and compactness were introduced . Metric and general topological spaces evolved in order to provide a framework in which to treat these types of convergence. As general topology gestated, many concepts were greatly clarified and simplified. (For example, “continuous” meant transforming convergent sequences into convergent sequences until about 1935.) These clarifications led to the development of general topological vector spaces in the 1930’s.