  We consider a nonparametric regression setup, where the covariate is a random
element in a complete separable metric space, and the parameter of interest
associated with the conditional distribution of the response lies in a
separable Banach space. We derive the optimum convergence rate for the kernel
estimate of the parameter in this setup. The small ball probability in the
covariate space plays a critical role in determining the asymptotic variance of
kernel estimates. Unlike the case of finite dimensional covariates, we show
that the asymptotic orders of the bias and the variance of the estimate
achieving the optimum convergence rate may be different for infinite
dimensional covariates. Also, the bandwidth, which balances the bias and the
variance, may lead to an estimate with suboptimal mean square error for
infinite dimensional covariates. We describe a data-driven adaptive choice of
the bandwidth, and derive the asymptotic behavior of the adaptive estimate.
