  A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold
$M$ endowed with a multiplication map $\mu\colon M \times M \to M$ such that
each left multiplication map $\mu_x := \mu(x,\cdot)$ (with $x \in M$) is an
involutive automorphism of $(M,\mu)$ with the isolated fixed point $x$. We show
that morphisms of Lie triple systems of symmetric spaces can be uniquely
integrated provided the first manifold is 1-connected. The problem is attacked
by showing that a continuous linear map between tangent spaces of affine Banach
manifolds with parallel torsion and curvature is integrable to an affine map if
it intertwines the torsion and curvature tensors provided the first manifold is
1-connected and the second one is geodesically complete. Further, we show that
the automorphism group of a connected Banach symmetric space $M$ can be turned
into a Banach-Lie group acting smoothly and transitively on $M$. In particular,
$M$ is a Banach homogeneous space.
