  This tutorial gives an elementary and self-contained review of abelian
bosonization in 1 dimension in a system of finite size $L$, following and
simplifying Haldane's constructive approach. As a non-trivial application, we
rigorously resolve (following Furusaki) a recent controversy regarding the
tunneling density of states, $\rho_{dos} (\omega)$, at the site of an impurity
in a Tomonaga-Luttinger liquid: we use finite-size refermionization to show
exactly that for g=1/2 its asymptotic low-energy behavior is
$\rho_{dos}(\omega) \sim \omega$. This agrees with the results of Fabrizio &
Gogolin and of Furusaki, but not with those of Oreg and Finkel'stein (probably
because we capture effects not included in their mean-field treatment of the
Coulomb gas that they obtained by an exact mapping; their treatment of
anti-commutation relations in this mapping is correct, however, contrary to
recent suggestions in the literature). --- The tutorial is addressed to readers
unfamiliar with bosonization, or for those interested in seeing ``all the
details'' explicitly; it requires knowledge of second quantization only, not of
field theory. At the same time, we hope that experts too might find useful our
explicit treatment of certain subtleties -- these include the proper treatment
of the so-called Klein factors that act as fermion-number ladder operators (and
also ensure the anti-commutation of different species of fermion fields), the
retention of terms of order 1/L, and a novel, rigorous formulation of
finite-size refermionization of both $e^{-i \Phi(x)}$ and the boson field $\Phi
(x)$ itself.
