Abstract: In this article, the bounded law of iterated logarithm for Banach space valued variables is discussed and four theorems are pregented. In particular, by some exponential inequalities Rainer Wittmann’s result is extended, i. e., that bounded law of the iterated logarithm holds for a sequence of independent, zero mean B-valued random variables V_n is shown provided （i） S_n/a_n is stochastically bounded, （ii） \sum_n=1^\infty a_n^-(2+\alpha) E\left\|V_n\right\|^2+\alpha<\infty \quad for 0 \leqslant \alpha \leqslant 1 (iii) \lim B_n=\infty, where S_n=\sum_i=1^n V_i, B_n=\left(\sum_j=1^n E\left\|V_l\right\|^2\right)^1 / 2, a_n=\left(2 B_n^2 \log _a B_n^2\right)^1 / 3 .