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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.29 Function Spaces

Terence Tao

1 What Is a Function Space?

When one works with real or complex numbers, there is a natural notion of the magnitude of a number x, namely its modulus |x|. One can also use this notion of magnitude to define a distance |x - y| between two numbers x and y and thereby say in a quantitative way which pairs of numbers are close and which ones are far apart.

The situation becomes more complicated, however, when one deals with objects with more degrees of freedom. Consider for instance the problem of determining the “magnitude” of a three-dimensional rectangular box. There are several candidates for such a magnitude: length, width, height, volume, surface area, diameter (the length of a long diagonal), eccentricity, ...

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