## Subspaces and projections

Let $M$ be a closed subspace of a Banach space $X$. In general, there is no linear projection $P\colon X\to M$, the canonical example being $c_0$ in $\ell^\infty$. At least we can construct a projection when $M$ is finite-dimensional. The one-dimensional case is easy: take a unit vector $x_1\in M$, pick a norming functional $f$ and define $P(x)=f(x)x_1$. If $M$ is $n$-dimensional, one can construct $P$ as the sum of $n$ rank-one projections, achieving $\|P\| \le n$. Which is pretty bad: distorting distances by a factor comparable to dimension may render high-dimensional data useless. One usually seeks estimates that are logarithmic in dimension (or better yet, dimension-independent).

Recall that a retraction is a continuous map $f\colon X\to M$ which is the identity on $M$. A projection is a linear retraction. The linearity is quite a rigid condition. We may have better luck with retractions in other classes, such as Lipschitz maps. And indeed, there is a 2-Lipschitz retraction from $\ell^\infty$ onto $c_0$. Given $x\in \ell^\infty$, let $d=\limsup|x_n|$ and define $r(x)=y$ with $y_n=(|x_n|-d)^+\mathrm{sign}\,x_n$. Since $d$ is a 1-Lipschitz function, $r$ is 2-Lipschitz. It is a retraction because $d=0$ when $x\in c_0$.

One of many open problems in the book Geometric Nonlinear Functional Analysis I (Benyamini and Lindenstrauss) is whether every Banach space is a Lipschitz retract of its bidual. This problem is also mentioned in 2007 survey by Nigel Kalton.

One thing to like about linear projections is their openness: any linear surjection between Banach spaces is an open map. This is not the case for Lipschitz surjections: for instance, $f(x)=(|x|-1)^+\mathrm{sign}\,x$ is a Lipschitz surjection $\mathbb R\to\mathbb R$ which maps $(-1,1)$ to a point. This example resembles the retraction $r$ above. And indeed, $r$ is not open either: the image of a small open ball centered at $(0,1,1,1,1,\dots)$ is contained in the hyperplane $x_1=0$.

In the context of Lipschitz maps it is natural to quantify openness in the same way as continuity: i.e., by requiring the image of a ball $B(x,r)$ to contain $B(f(x),r/C)$ with $C$ independent of $x,r$. This defines Lipschitz quotients, which appear to be the right concept of “nonlinear projection”. However, it remains unknown whether there is a Lipschitz quotient $Q\colon \ell^\infty\to c_0$. [Benyamini and Lindenstrauss]

## Compactness of operators, and a lot of projections

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space. Claim: an operator $T\colon\mathcal{H}\to\mathcal{H}$ is compact if and only if $Te_n\to 0$ for every orthonormal sequence $\lbrace e_n\rbrace$.
Continue reading “Compactness of operators, and a lot of projections”