Square variation of Brownian paths in Banach spaces - Mou-Hsiung Chang (International Journal of Mathematics and Mathematical Sciences)
It is known that if {W(t),   0≤t≤1} is a standard Brownian motion in ℝ then limn→∞∑i=12n|W(i/2n)−W((i−1)/2n)|2=1 almost surely. We generalize this celebrated theorem of Levy to Brownian motion in real separable Banach spaces.