ABSTRACT. Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result.Introduction. Let H denote a real separable Hilbert space with norm ||-||w and assume ||-||B is a measurable norm on H in the sense of [2]. Then there exists a constant M > 0 such that ||x||¿ < M||x||w for all x G H, and if B is the completion of H in ||-|la it follows that B is a real separable Banach space. We will view H as a subspace of B and since ||-||a is weaker than \\-\\H on H it follows that B*, the topological dual of B, can be continuously injected into //*, the topological dual of H. We call (H, B) an abstract Wiener space.For t > 0, let m, denote the canonical Gaussian cylinder set measure on H with variance parameter t and let ft» (t > 0) denote the Borel probability measure on B induced by m, (t > 0). We call ft, the Wiener measure on B generated by H with variance parameter t.Let flB denote the space of continuous functions w from [0, oo) into B such that w(0) = 0, and let D be the o-field of fiB generated by the functions w -» w(t). Then there is a unique probability measure P on D such that if 0 = t0 < tx <•••<'" then w(tj) -w(tj_x) (j = l,...,n) are independent and w(tj) -w(tj_x) has distribution ft,y_, | on B. In particular, the stochastic process W¡ defined on (fifl,0,F) by W¡(w) = w(t) has stationary independent Gaussian