Embeddings II: searching for roundness in ugliness

The concluding observation of Part I was that it’s hard to embed things into a Hilbert space: the geometry is the same in all directions, and the length of diagonals of parallelepipeds is tightly controlled. One may think that it should be easier to embed the Hilbert space itself into other things. And this is indeed so.

Let’s prove that the space L^p[0,1] contains an isomorphic copy of the Hilbert space \ell_2, for every 1\le p \le \infty. The case p=\infty can be immediately dismissed: this is a huge space which, by virtue of Kuratowski’s embedding, contains an isometric copy of every separable metric space. Assume 1\le p<\infty.

Recall the Rademacher functions r_n(t)=\mathrm{sign}\, \sin (2^{n+1}\pi t), n=0,1,2,3,\dots They are simply square waves:

Define a linear operator T\colon \ell_2\to L^p[0,1] by T(c_1,c_2,\dots)=\sum_{n}c_nr_n. Why is the sum \sum_{n}c_nr_n in L^p, you ask? By the Хинчин inequality:

\displaystyle A_p\sqrt{\sum c_n^2} \le \left\|\sum c_n r_n\right\|_{L^p} \le B_p\sqrt{\sum c_n^2}

The inequality tells us precisely that T is an isomorphism onto its image.

So, even the indescribably ugly space L^1[0,1] contains a nice roundish subspace.
(Why is L^1[0,1] ugly? Every point of its unit sphere is the midpoint of a line segment that lies on the sphere. Imagine that.) One might ask if the same holds for every Banach space, but that’s way too much to ask. For instance, the sequence space \ell_p for p\ne 2,\infty does not have any subspace isomorphic to \ell_2. Informally, this is because the underlying measure (the counting measure on \mathbb N) is not infinitely divisible; the space consists of atoms. For any given N we can model the first N Rademacher functions on sequences, but the process has to stop once we reach the atomic level. On the positive side, this shows that \ell_p contains isomorphic copies of Euclidean spaces of arbitrarily high dimension, with uniform control on the distortion expressed by A_p and B_p. And this property is indeed shared by all Banach spaces: see Dvoretzky’s theorem.

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