Using homological methods on the base of iterated spectra in functional analysis

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or $H$-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular $H$-limit are projective and inductive limits of separated locally convex spaces. The class of $H$-spaces contains Fréchet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quotient spaces. Moreover, for $H$-space an unproved variant of the closed graph theorem holds true. Homological methods are used for proving of theorems of vanishing at zero for first derivative of Hausdorff limit functor: $\mathrm{Haus}^1(\boldsymbol X)=0$.