Let be a closed subspace of a Banach space . In general, there is no linear projection , the canonical example being in . At least we can construct a projection when is finite-dimensional. The one-dimensional case is easy: take a unit vector , pick a norming functional and define . If is -dimensional, one can construct as the sum of rank-one projections, achieving . 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 which is the identity on . 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 onto . Given , let and define with . Since is a 1-Lipschitz function, is 2-Lipschitz. It is a retraction because when .
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, is a Lipschitz surjection which maps to a point. This example resembles the retraction above. And indeed, is not open either: the image of a small open ball centered at is contained in the hyperplane .
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 to contain with independent of . This defines Lipschitz quotients, which appear to be the right concept of “nonlinear projection”. However, it remains unknown whether there is a Lipschitz quotient . [Benyamini and Lindenstrauss]