# On the ${\Ext}^2$-problem for Hilbert spaces

## TL;DR

This paper proves that the second Ext group for certain Banach and quasi-Banach spaces is non-zero, resolving longstanding problems related to extensions and local convexity in functional analysis.

## Contribution

It demonstrates that xt^2(\u2113_2, xt_2) and xt^2(ll_1, \u2126) are non-zero, solving key problems in Banach space theory and quasi-Banach space extensions.

## Key findings

- xt^2(ll_2, xt_2) 0 in Banach spaces
- xt^2(ll_1, ) 0 in quasi Banach spaces
- Solves the second order Palais problem and four-space problem for local convexity.

## Abstract

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.