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?

  • 7
    $\begingroup$ 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. $\endgroup$ Sep 3, 2019 at 5:00
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    $\begingroup$ 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. $\endgroup$ Sep 3, 2019 at 20:16


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