Development of the Fundamental Notions of Functional Analysis
Overview
In the early twentieth century, mathematicians learned to give a geometrical interpretation to sets of functions that met certain overall conditions. This interpretation allowed mathematicians to assign a norm or "length" to each function in the set and to provide a measure of how much two functions differed from each other. One set of conditions on the functions in a set defined a Hilbert space, which could be treated as a vector space of infinite dimensions. Somewhat more general conditions were allowed in a Banach space. Both types of function spaces are of great importance in modern applied mathematics, and the ideas of Hilbert space play a particularly important role in quantum physics.
Background
At the close of the nineteenth century many mathematicians believed that the theory of sets,introduced by the Russian-born German mathematician Georg Cantor (1845-1918), would provide a reliable and unifying basis for all mathematics. While not all mathematicians shared this view, it was embraced and popularized by the great German mathematician, David Hilbert (1862-1943), who in 1926 wrote, "No one shall expel us from the paradise which Cantor has created for us." By that time a small army of mathematicians had been at work, establishing a rigorous basis for set theory itself and providing set theoretic interpretations for algebra, geometry, the calculus, and the theory of numbers.
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