On nonseparable Banach spaces

Abstract:

Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of ${L^1}{\{ - 1, 1\} ^\alpha }$ into ${X^{\ast }}$ implies, for any infinite cardinal $\alpha$, the isomorphic embedding of $l_\alpha ^1$ into $X$, gets the following form: if $\alpha = \omega$, has been proved from Pełczynski; if $\alpha > {\omega ^ + }$, the proof is given in this paper; if $\alpha = {\omega ^ + }$, in ${\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + {\text {C}}{\text {.H}}{\text {.}}$, an example discovered by Haydon gives a negative answer; if $\alpha = {\omega ^ + }$, in ${\text {Z}}{\text {.F}}{\text {.C}}{\text {.}} + \urcorner {\text {C}}{\text {.H}}{\text {.}} + {\text {M}}{\text {.A}}{\text {.}}$, is also proved in this paper.