Spaces and singular perturbations on manifolds without boundary Singular perturbations one of the central topics in asymptotic analysis also play a special role in describing physical phenomena such as the propagation of waves in media in the presence of small energy dissipations or dispersions the appearance of boundary or interior layers in fluid and gas dynamics as well as in elasticity theory semi classical asymptotic approximations in quantum mechanics etc. Elliptic and coercive singular perturbations are of special interest for the asymptotic solution of problems which are characterized by boundary layer phenomena e.g. the theory of thin buckling plates elastic rods and beams. This first volume deals with linear singular perturbations (on smooth manifolds without boundary) considered as equicontinuous linear mappings between corresponding families of Sobolev Slobodetskis type spaces of vectorial order