Waj. Luxemburg et M. Vath, The existence of non-trivial bounded functionals implies the Hahn-Banach extension theorem, Z ANAL ANWE, 20(2), 2001, pp. 267-279
We show that it is impossible to prove the existence of a linear (bounded o
r unbounded) functional on any L-infinity/C-o without an uncountable form o
ft he axiom of choice. Moreover., we show that if on each Banach space ther
e exists at least one non-trivial bounded linear functional, then the Hahn-
Banach extension theorem must hold. We also discuss relations of non-measur
able sets and the Hahn-Banach extension theorem.