  The importance of Schur complement based preconditioners are well-established
for classical saddle point problems in $\mathbb{R}^N \times \mathbb{R}^M$. In
this paper we extend these results to multiple saddle point problems in Hilbert
spaces $X_1\times X_2 \times \cdots \times X_n$. For such problems with a block
tridiagonal Hessian and a well-defined sequence of associated Schur
complements, sharp bounds for the condition number of the problem are derived
which do not depend on the involved operators. These bounds can be expressed in
terms of the roots of the difference of two Chebyshev polynomials of the second
kind. If applied to specific classes of optimal control problems the abstract
analysis leads to new existence results as well as to the construction of
efficient preconditioners for the associated discretized optimality systems.
