Let $J$ be the James space. I have the following questions:


Question 1: Does every infinite-dimensional closed subspace of $J$ contain an infinite-dimensional closed subspace that is $C$-complemented in $J$? where the $C$ is the universal constant.


Question 2: Let $(u\_{n})\_{n}$ be a normalized skipped block basic sequence of the unit vector basis in $J$. Is the subspace spanned by $(u\_{n})\_{n}$ $C$-complemented in $J$? where the $C$ is the universal constant.


Obviously, if Question 2 is true, then Question 1 is true. I do not know what is the above universal constant $C$.


Thank you!



Complemented subspaces in the James space
The answer to both questions is **yes**. Indeed, suppose $y\_j=\sum\_{n=p\_j}^{q\_j}\alpha\_ne\_n$ forms a block basic sequence in $J$ satisfying $p\_{j+1}-q\_j>1$ for all $j$. It is shown in the proof of Theorem 2.d.2 of *The James Forest* that $[y\_j]\_{j=1}^\infty$ is complemented by a projection of norm $\leq 2\sqrt{2}$. (It is also shown that if $(y\_i)\_{i=1}^\infty$ is seminormalized then it is equivalent to the $\ell\_2$ basis.)


