×

On the computation of minimal projections: Millenium report. (English) Zbl 0970.46007

Anastassiou, George A. (ed.), Applied mathematics reviews. Volume 1. Singapore: World Scientific. 119-156 (2000).
This is a most interesting survey. Particularly so because the authors have gone to great lengths to make it easily accessible and interesting to a general mathematical readership. The introduction sets out the problem with great clarity. The rest of this review consists of (modifed) excerpts from the introduction. Let \(X\) be a Banach space and \(V\) a non-trivial \(n\)-dimensional subspace of \(X\). There are infinitely many projections \(P\) of \(X\) onto \(V\). Different projections may have different norms. A projection is minimal if \[ \|P\|= \lambda(V,X):=\inf\{\|Q\|: Q \text{ a projection of \(X\) onto \(V\)}\}. \] The general problem is to construct a minimal projection. If \((v_1,v_2,\ldots,v_n)\) is a basis for \(V\) then a general projection onto \(V\) has the form \[ P = \sum_{j=1}^n u_j\otimes v_j, \text{\quad i.e. } Px = \sum_{j=1}^n u_j (x)v_j \] where \(u_j \in X^*\) are linear functionals such that \(u_j(v_k) = \delta_{jk}\). One can reformulate the problem as an optimization problem: given a basis \((v_j)\) for \(V\), minimize \(\|\sum u_j\otimes v_j\|\) with constraints \(u_j \in X^*\) and \(u_j(v_k) = \delta_{jk}.\) It turns out that in many situations the number \(\lambda(V,X)\) is an isometric invariant; i.e., if \(V_1\) and \(V_2\) are isometric then \(\lambda(V_1,X) = \lambda(V_2,X)\). Thus, in these cases, it depends only on the shape of the ball in \(V\). In some cases, even the form of the minimal projections is an isometric invariant. The authors discuss the progress that has been made on the problem. They also give many examples and applications. They point out that the previously known examples “are the precise exceptions. They are unlike what is to be expected from the general form of minimal projections”.
For the entire collection see [Zbl 0945.00005].

MSC:

46B20 Geometry and structure of normed linear spaces
PDFBibTeX XMLCite