[Paper Review] Banach spaces with no proximinal subspaces of codimension 2

C. J. Read|arXiv (Cornell University)|Jul 30, 2013

Advanced Banach Space Theory1 references4 citations

TL;DR

This paper constructs a Banach space $(c_0, \|\cdot\|)$ with a carefully designed norm to demonstrate that not all Banach spaces admit proximinal subspaces of finite codimension $n \geq 2$. By leveraging a sequence of rational sequences and sparse perturbations via $\ell^1$-functionals, the authors show that the Gâteaux derivative structure of the norm prevents the existence of minimal-distance projections to any finite-codimensional subspace, thereby providing a counterexample to a long-standing conjecture by Ivan Singer.

ABSTRACT

The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving functionals have to be: if (x,f) is in X x X* with ||x||=||f||=1 and |1-f(x)|< h^2/4 then there are (x',f') in X x X* with ||x'||= ||f'||=1, ||x-x'||, ||f-f'||< h and f'(x')=1. This means that there are always "proximinal" hyperplanes H in X (a nonempty subset E of a metric space is said to be "proximinal" if, for x not in E, the distance d(x,E) is always achieved - there is always an e in E with d(x,E)=d(x,e)); for if H= ker f (f in X*) then it is easy to see that H is proximinal if and only if f is norm-achieving. Indeed the set of proximinal hyperplanes H is, in the appropriate sense, dense in the set of all closed hyperplanes H in X. Quite a long time ago [Problem 2.1 in his monograph "The Theory of Best approximation and Functional Analysis" Regional Conference series in Applied Mathematics, SIAM, 1974], Ivan Singer asked if this result generalized to closed subspaces of finite codimension - if every Banach space has a proximinal subspace of codimension 2, for example. In this paper I show that there is a Banach space X such that X has no proximinal subspace of finite codimension n>1. So we have a converse to Bishop-Phelps-Bollobas: a dense set of proximinal hyperplanes can always be found, but proximinal subspaces of larger, finite codimension need not be.

Motivation & Objective

• To resolve a long-standing open problem posed by Ivan Singer regarding the existence of proximinal subspaces of finite codimension in Banach spaces.
• To demonstrate that while proximinal hyperplanes (codimension 1) are always dense by the Bishop-Phelps-Bollobás theorem, this fails for codimension ≥2.
• To construct a specific Banach space $(c_0, \|\cdot\|)$ with a non-standard norm that blocks proximinality in finite codimension ≥2.
• To show that the geometric and analytic structure of the norm—based on sparse, rational perturbations—prevents the existence of minimal projections to finite-codimensional subspaces.
• To provide a sharp contrast between the density of proximinal hyperplanes and the nonexistence of such subspaces in higher codimensions.

Proposed method

• Define a new norm on $c_0$ by adding weighted $\ell^1$-seminorms: $\|x\| = \|x\|_0 + \sum_{k=1}^\infty 2^{-a_k^2} |\langle x, u_k - e_{a_k} \rangle|$, where $u_k \in c_{00}(\mathbb{Q})$ and $a_k$ satisfy growth conditions.
• Ensure the perturbation terms are sparse and grow rapidly so that $\|x\| \leq 3\|x\|_0$, preserving the $c_0$ structure while introducing directional non-smoothness.
• Use Gâteaux derivative analysis to study the norm's differentiability and exploit the lack of directional smoothness at points in finite-codimensional subspaces.
• Construct a finite sequence of functionals $\psi_r$ and approximate them by $\ell^1$-elements $z_r$ to induce sign patterns on inner products with candidate minimal vectors.
• Apply a quotient map $\theta: \Phi_0 \to \ell^\infty(A)/c_0(A)$ to analyze the image of the functional space $\Phi_0$, showing it must contain $N+1$ linearly independent vectors.
• Derive a contradiction by showing $\dim \theta\Phi_0 \geq N+1$ while $\dim \Phi_0 = N$, proving that no such proximinal subspace $H$ of codimension $N \geq 2$ can exist.

Experimental results

Research questions

• RQ1 Does every Banach space admit a proximinal subspace of finite codimension $n \geq 2$?
• RQ2 Can the classical density result for proximinal hyperplanes (Bishop-Phelps-Bollobás) be extended to higher codimensions?
• RQ3 Is there a Banach space where no finite-codimensional subspace of codimension $n \geq 2$ is proximinal?
• RQ4 What structural properties of a norm prevent proximinality in finite codimension?
• RQ5 Can a non-smooth, sparse perturbation of the $c_0$ norm block proximinality beyond codimension 1?

Key findings

• The constructed Banach space $(c_0, \|\cdot\|)$ has no proximinal subspace of finite codimension $n \geq 2$, providing a counterexample to Singer's conjecture.
• The norm is defined via a sum of $\ell^1$-seminorms over a sparse, rapidly growing sequence $a_k$, ensuring $\|x\| \leq 3\|x\|_0$.
• The Gâteaux derivative of the norm fails to exist in a controlled way at points in finite-codimensional subspaces, preventing minimal projections.
• The image of the functional space $\Phi_0$ under the quotient map $\theta$ must have dimension at least $N+1$, contradicting $\dim \Phi = N$.
• The sign patterns of inner products with approximating functionals $z_r$ generate $N+1$ linearly independent vectors in $\mathbb{R}^{N+1}$, forcing the contradiction.
• The result establishes a sharp dichotomy: proximinal hyperplanes are always dense, but no such guarantee holds for codimension ≥2.