# Status of two Banach space theory open problems posted by Pełczyński

In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems.

Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is not isomorphic to a Hilbert space.

(i) (A. Pełczyński) Does there exist an infinite dimensional subspace of $X$ with Schauder basis that is not complemented in $X?$

(ii) (A. Pełczyński) Do there exist two infinite-dimensional subspaces of $X$ with Schauder basis that are not isomorphic?

I would like to know status of the two problems above. Are they solved or still open?

• I believe these are open (and I think they are equivalent to each other). However, if you relax the assumption a bit as `every subspace is isomorphic to a complemented subspace', then there are non-Hilbertian examples. These were constructed by Johnson-Szankowski math.tamu.edu/~bill.johnson/HAP5.2.4.pdf – Bunyamin Sari Apr 24 '18 at 18:36
• @BunyaminSari: Thanks. If possible, can you provide a reference showing that the two questions are equivalent? – Idonknow Apr 25 '18 at 1:05
• I think that it is even open whether $X$ must be isomorphic to a Hilbert space if every subspace of $X$ with a Schauder basis is complemented and isomorphic to a Hilbert space. – Bill Johnson Apr 26 '18 at 16:47