  We investigate if an existing notion of weak sequential convergence in a
Hadamard space can be induced by a topology. We provide an answer on what we
call weakly proper Hadamard spaces. A notion of dual space is proposed and it
is shown that our weak topology and dual space coincide with the standard ones
in the case of a Hilbert space. Moreover we introduce the space of geodesic
segments and a corresponding weak topology, and we show that this space is
homeomorphic to its underlying Hadamard space. As an application of it we show
the existence of a geodesic segment that acts as direction of steepest descent
for a geodesically differentiable function whose geodesic derivative satisfies
certain properties. Finally we extend several results from classical functional
analysis to the setting of Hadamard spaces, and we compare our topology with
other existing notions of weak topologies.
