A general bounded law of the iterated logarithm for Banach space value
d random variables is established. Our result implies: (a) the bounded
LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables i
n the domain of attraction of a Gaussian law and (c) new LIL results f
or random variables outside the domain of attraction of a Gaussian law
in cases where the classical norming sequence {root nLLn} does not wo
rk. Basic ingredients of our proof are an infinite-dimensional Fuk-Nag
aev type inequality and an infinite-dimensional version of Klass's K-f
unction.