Meaning of twisted sum

Question

When I was studying for my functional analysis final exam I came across the this Wikipedia article. I do not understand the concepts of F-space extensions nor twisted sum. Is there any bibliography recommended?

Answer 1

Suppose that you have a vector space $X$ and a linear subspace $Y$. Then you may always find a linear complement, that is, a linear subspace $Z$ such that $X = Y\oplus Z$.

When $X$ is a Banach space (or more generally, an F-space), for a given closed subspace $Y$ it may be impossible to find a closed subspace $Z$ such that $X=Y \oplus Z$. (For a concrete example take $X=\ell_\infty$ and $Y=c_0$.) However, we may always take the quotient $X/Y$ and call it $Z$.

Twisted (or snarked) sum of Banach spaces $Y$ and $Z$ is a Banach space $X$ such that $Y$ embeds into $X$ via some isomorphism $T$ and $Z\cong X/T(Y)$. This is the general philosophy and there is a big industry of identifying those properties of Banach spaces that pass to $X$ from $Y$ and $Z$. As for the literature, I would recommend

1. J.M.F. Castillo and M. González, Three-space problems in Banach space theory, Lecture Notes in Math. 1667, Springer-Verlag, 1997.
2. N.J. Kalton, N.T. Peck, J.W. Roberts, An F-space Sampler, London Math. Soc. Lecture Notes 89 (1984).