We  introduce  here  new  concepts  of  functional  analysis:  Hausdorff  spectrum  and Hausdorff  limit  or  H-limit  of Hausdorff  spectrum  of  locally  convex  spaces.  Author has introduced  this  concepts  in  2002  but  progress  in different areas of mathematics (algebraic geometry, differential equations, category theory, ets) defined the need to  expand  fundamental ...