Let $M$ be a finite-dimensional subspace of $c_{0}$, and let $\varepsilon>0$.

Question. Does there exist a finite rank projection from $c_{0}$, of norm $\leq 1+\varepsilon$, onto a subspace $N$ of $c_{0}$ with $M\subseteq N$ and Banach-Mazur distance $\textrm{d}(N,l_{\infty}^{n})\leq 1+\varepsilon$, where $n=\textrm{dim} N$ ?

  • $\begingroup$ Isn't this the same as asking if $X$ is a ${\mathcal L}_p$-space, or have I misread something? $\endgroup$
    – Yemon Choi
    Apr 10, 2021 at 4:01
  • $\begingroup$ @YemonChoi My question is sharper than $X$ is a $\mathcal{L}_{p}$-space. $\endgroup$ Apr 10, 2021 at 4:25
  • 2
    $\begingroup$ What is the difference between what your asking and that $c_0$ is an $\mathscr{L}_{\infty, 1+\varepsilon}$-space? The norm-1 projection onto $\ell_\infty^n$ is for free, isn't it? $\endgroup$ Apr 10, 2021 at 17:39
  • $\begingroup$ @DirkWerner The norm-1 projection onto $l_{\infty}^{n}$ is for free because $l_{\infty}^{n}$ is 1-injective. You are right. $\endgroup$ Apr 11, 2021 at 1:12

1 Answer 1


For non-zero sequence $x=(x_1, x_2, \ldots)$ denote $L(x)=\min\{i:x_i\ne 0\}$ (leader of $x$). By Gauss elimination, $M$ contains a basis $(p_1, \ldots, p_d) $ with distinct leaders $m_1<m_2<\ldots<m_d$ respectively. Let $f_1, f_2, \ldots$ denote consecutive standard basic vectors $e_k$ with $k\notin \{m_1,\ldots,m_d\}$. Consider the span $N$ of $M$ and $f_1, \ldots, f_s$ with very large $s$. On this $(d+s)$-dimensional space $N$ with the natural basis $\{p_1,\ldots,p_d,f_1,\ldots,f_s\}$, the coordinate functionals corresponding to $p_i$'s are uniformly (by $s$) bounded: actually their expressions via the usual coordinate functionals do not depend on $s$. Now you cut all $p_j$'s on the corresponding level $d+s$: denote $n_i=P_{d+s} p_i$ where $P_{d+s} x=(x_1,\ldots,x_{d+s},0,0,\ldots)$. The span $N$ of $n_1,\ldots,n_d,f_1,\ldots,f_s$ is the coordinate subspace $l_\infty^{d+s}\subset c_0$. The map $\Phi\colon M\to N$ which acts as $$\Phi\left( \alpha_1p_1+\ldots+\alpha_sp_s+\beta_1f_1+\ldots+\beta_s f_s\right)= \alpha_1n_1+\ldots+\alpha_sn_s+\beta_1f_1+\ldots+\beta_s f_s$$ satisfies $\|x-\Phi x\|<\varepsilon_s$ for all $x$ on the unit sphere of $M$ and $\varepsilon_s$ tending to 0 for large $s$ (since $\alpha_i's$ are uniformly bounded as observed above, and $n_i$'s become close to $p_i$'s when $s$ is large). This proves that $N$ and $M$ are Banach -- Mazur close.

The existence of a norm-1 projection to $l_\infty^n$ is a general fact, as observed by Dirk Werner in the comments.

  • $\begingroup$ Thank you for a nice proof, Fedor. $\endgroup$ Apr 12, 2021 at 0:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.