Near cohomologies of Hilbert complexes are obtained heuristically by taking cochains with small coboundaries modulo cochains which are close to cocycles. Rigorously this leads to a family of closed cones depending on a small real parameter up to an equivalence relation. It is proved that the near cohomologies are homotopy invariants of a Hilbert complex with respect to the chain homotopy equivalence defined by morphism and homotopy operators which are bounded linear operators. Applying this to the Hilbert de Rham complex on the universal covering of a non-simply connected manifold gives homotopy invariants of this manifold. A von Neumann algebra stucture on a Hilbert complex allows to convert near-cohomologies to number homotopy invariants of the complex. For the Hilbert de Rham complex they coincide with the invariants introduced and investigated by the authors in an earlier paper and include heat kernel decay exponents by S.P. Novikov and M.A. Shubin.