  In this paper, we prove that functional sliced inverse regression (FSIR)
achieves the optimal (minimax) rate for estimating the central space in
functional sufficient dimension reduction problems. First, we provide a
concentration inequality for the FSIR estimator of the covariance of the
conditional mean, i.e., $\var(\E[\boldsymbol{X}\mid Y])$. Based on this
inequality, we establish the root-$n$ consistency of the FSIR estimator of the
image of $\var(\E[\boldsymbol{X}\mid Y])$. Second, we apply the most widely
used truncated scheme to estimate the inverse of the covariance operator and
identify
  the truncation parameter which ensures
  that FSIR can achieve the optimal minimax convergence rate for estimating the
central space. Finally, we conduct simulations to demonstrate the optimal
choice of truncation parameter and the estimation efficiency of FSIR. To the
best of our knowledge, this is the first paper to rigorously prove the minimax
optimality of FSIR in estimating the central space for multiple-index models
and general $Y$ (not necessarily discrete).
