  Due to simplicity, computational cheapness, and efficiency, the Barzilai and
Borwein (BB) gradient method has received a significant amount of attention in
different fields of optimization. In the first part of this paper, based on
spectral analysis, R-linear global convergence for the BB-method is proven for
strictly convex quadratic problems posed in infinite-dimensional Hilbert
spaces. Then this result is strengthened to R-linear local convergence for a
class of twice continuously Fr\'echet-differentiable functions. In the second
part, aiming at problems governed by partial differential equations (PDE), the
mesh-independent principle is investigated for the BB-method. The applicability
of these results is demonstrated for three different types of PDE-constrained
optimization problems. Numerical experiments illustrate the theoretical
results.
