# Finite-dimensional subspaces of $c_{0}$

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 $$n=\textrm{dim} N$$ ?

• Isn't this the same as asking if $X$ is a ${\mathcal L}_p$-space, or have I misread something? Apr 10, 2021 at 4:01
• @YemonChoi My question is sharper than $X$ is a $\mathcal{L}_{p}$-space. Apr 10, 2021 at 4:25
• 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? Apr 10, 2021 at 17:39
• @DirkWerner The norm-1 projection onto $l_{\infty}^{n}$ is for free because $l_{\infty}^{n}$ is 1-injective. You are right. Apr 11, 2021 at 1:12

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 $$m_1 respectively. Let $$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.