Distance between functions
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When Fréchet and Hausdorff introduced the notions of distance and metric space (see the article "The genesis of spaces"), it became clear that these notions would apply to function spaces. Here too, as we saw for the plane (in the article "Balls in the plane") and space (in the article "Surprising shapes in space"), distances most often arise from norms. The study of these spaces by many mathematicians, including the German mathematician David Hilbert and the Hungarian mathematician Frigyes Riesz (1880–1956, not to be confused with his brother Marcel), led to the concept of a Banach space, named after the Polish mathematician Stefan Banach (1892–1945), who studied them with others in the early 1920s.
A Banach space is simply a normed vector space in which, much as with the real numbers, there is a relatively simple criterion for showing that a sequence of elements converges… without necessarily having to know its limit.
When balls look like tubes
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One of the simplest norms used in Banach-space theory is defined on a space of real-valued continuous functions. Consider, for example, functions defined on the interval [0, 1]. Let E = C ([0, 1], ℝ). For each f* in E, define the following non-negative number:
