  We investigate the classical stability of the higher-dimensional
Schwarzschild black holes against linear perturbations, in the framework of a
gauge-invariant formalism for gravitational perturbations of maximally
symmetric black holes, recently developed by the authors. The perturbations are
classified into the tensor, vector, and scalar-type modes according to their
tensorial behaviour on the spherical section of the background metric, where
the last two modes correspond respectively to the axial- and the polar-mode in
the four-dimensional situation. We show that, for each mode of the
perturbations, the spatial derivative part of the master equation is a
positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the
master equation for each tensorial type of perturbations does not admit
normalisable negative-modes which would describe unstable solutions.
  On the same Schwarzschild background, we also analyse the static perturbation
of the scalar mode, and show that there exists no static perturbation which is
regular everywhere outside the event horizon and well-behaved at spatial
infinity. This checks the uniqueness of the higher-dimensional spherically
symmetric, static, vacuum black hole, within the perturbation framework.
  Our strategy for the stability problem is also applicable to the other
higher-dimensional maximally symmetric black holes with non-vanishing
cosmological constant. We show that all possible types of maximally symmetric
black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and
Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the
vector perturbations.
