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.

The Khintchine inequality

Today’s technology should make it possible to use the native transcription of names like Хинчин without inventing numerous ugly transliterations. The inequality is extremely useful in both analysis and probability: it says that the L^p norm of a linear combination of Rademacher functions (see my post on the Walsh basis) can be computed from its coefficients, up to a multiplicative error that depends only on p. Best of all, this works even for the troublesome p=1; in fact for all 0<p<\infty. Formally stated, the inequality is

\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}

where the constants A_p,B_p depend only on p. The orthogonality of Rademacher functions tells us that A_2=B_2=1, but what are the other constants? They were not found until almost 60 years after the inequality was proved. The precise values, established by Haagerup in 1982, behave in a somewhat unexpected way. Actually, only A_p does. The upper bound is reasonably simple:

\displaystyle B_p=\begin{cases} 1, \qquad 0<p\le 2 \\ \sqrt{2}\left[\Gamma(\frac{p+1}{2})/\sqrt{\pi}\right]^{1/p},  \qquad 2<p<\infty \end{cases}

The lower bound takes an unexpected turn:

\displaystyle A_p=\begin{cases} 2^{\frac{1}{2}-\frac{1}{p}},\qquad 0<p\le p_0 \\  \sqrt{2}\left[\Gamma(\frac{p+1}{2})/\sqrt{\pi}\right]^{1/p}, \qquad p_0<p<2 \\  1,\qquad 2\le p<\infty \end{cases}

The value of p_0 is determined by the continuity of A_p, and is not far from 2: precisely, p_0\approx 1.84742. Looks like a bug in the design of the Universe.

Rademacher series

For a concrete example, I took random coefficients c_0...c_4 and formed the linear combination shown above. Then computed its L^p norm and the bounds in the Khintchine inequality. The norm is in red, the lower bound is green, the upper bound is yellow.

Two-sided bounds

It’s a tight squeeze near p=2