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From Tim Gowers, I hear the good news that the editing process of the Princeton Companion to Mathematics is finally nearing completion. It therefore seems like a good time to resume my own series of Companion articles, while there is still time to correct any errors.

I’ll start today with my article on “Function spaces“. Just as the analysis of numerical quantities relies heavily on the concept of magnitude or absolute value to measure the size of such quantities, or the extent to which two such quantities are close to each other, the analysis of functions relies on the concept of a norm to measure various “sizes” of such functions, as well as the extent to which two functions resemble to each other. But while numbers mainly have just one notion of magnitude (not counting the p-adic valuations, which are of importance in number theory), functions have a wide variety of such magnitudes, such as “height” ($L^\infty$ or $C^0$ norm), “mass” ($L^1$ norm), “mean square” or “energy” ($L^2$ or $H^1$ norms), “slope” (Lipschitz or $C^1$ norms), and so forth. In modern mathematics, we use the framework of function spaces to understand the properties of functions and their magnitudes; they provide a precise and rigorous way to formalise such “fuzzy” notions as a function being tall, thin, flat, smooth, oscillating, etc. In this article I focus primarily on the analytic aspects of these function spaces (inequalities, interpolation, etc.), leaving aside the algebraic aspects or the connections with mathematical physics.

The Companion has several short articles describing specific landmark achievements in mathematics. For instance, here is Peter Cameron‘s short article on “Gödel’s theorem“, on what is arguably one of the most popularised (and most misunderstood) theorems in all of mathematics.

Having studied compact extensions in the previous lecture, we now consider the opposite type of extension, namely that of a weakly mixing extension. Just as compact extensions are “relative” versions of compact systems, weakly mixing extensions are “relative” versions of weakly mixing systems, in which the underlying algebra of scalars ${\Bbb C}$ is replaced by $L^\infty(Y)$. As in the case of unconditionally weakly mixing systems, we will be able to use the van der Corput lemma to neglect “conditionally weakly mixing” functions, thus allowing us to lift the uniform multiple recurrence property (UMR) from a system to any weakly mixing extension of that system.

To finish the proof of the Furstenberg recurrence theorem requires two more steps. One is a relative version of the dichotomy between mixing and compactness: if a system is not weakly mixing relative to some factor, then that factor has a non-trivial compact extension. This will be accomplished using the theory of conditional Hilbert-Schmidt operators in this lecture. Finally, we need the (easy) result that the UMR property is preserved under limits of chains; this will be accomplished in the next lecture.