  In this paper we give some basic and important properties of several typical
Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear
expectation--G-expectation. Many results can be also applied to more general
situations. A generalized version of Kolmogorov's criterion for continuous
modification of a stochastic process is also obtained. The results can be
applied to continuous time dynamic and coherent risk measures in finance in
particular for path-dependence risky positions under situations of volatility
model uncertainty.
