  This paper is concerned with the convergence analysis of two-level
hierarchical basis (TLHB) methods in a general setting, where the decomposition
associated with two hierarchical component spaces is not required to be a
direct sum. The TLHB scheme can be regarded as a combination of compatible
relaxation and coarse-grid correction. Most of the previous works focus on the
case of exact coarse solver, and the existing identity for the convergence
factor of exact TLHB methods involves a tricky max-min problem. In this work,
we present a new and purely algebraic analysis of TLHB methods, which gives a
succinct identity for the convergence factor of exact TLHB methods. The new
identity can be conveniently utilized to derive an optimal interpolation and
analyze the influence of coarse space on the convergence factor. Moreover, we
establish two-sided bounds for the convergence factor of TLHB methods with
inexact coarse solver, which extend the existing TLHB theory.
