# Complementability of finite dimensional subspaces

Suppose $$X$$ is a separable infinite dimensional Banach space, $$E$$ a finite dimensional subspace which is $$c$$-complemented in $$X$$. Is the following true?

For any $$\varepsilon>0$$, one can find $$x\notin E$$ such that $$\mathrm{span}\{E,x\}$$ is $$(c+\varepsilon)$$-complemented in $$X$$.

This is obviously true in Hilbert spaces, but not sure in general Banach spaces. Would reflexivity make a difference?

• This is not true because, if we use this condition repeatedly, we would get that there exist subspaces in $X$ with arbitrarily large dimension which are $1+\epsilon$-complemented. This is known to be false in a very strong sense, see G. Pisier, Acta Math. 151 (1983), no. 3-4, 181–208; or his book "Factorization of Linear Operators", 1986, Chapter 10, Corollary 10.8. Pisier mentions that (in 1986) it was unknown whether such spaces can be reflexive, and I do not know whether this question was solved. Sep 3, 2019 at 5:00
• It is a nice elementary exercise to prove this is true if $X$ is an $L_p$ space, $1\le p < \infty$. I think I will assign this problem to the students in my functional analysis class. No partial credit for doing the $p=2$ case. Sep 3, 2019 at 20:16