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]

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