We take another approach to the well known
theorem of Korovkin, in the following situation:
$X$, $Y$ are compact Hausdorff spaces,
$M$ is a unital subspace of
the Banach space $C(X)$ (respectively, $C_{{\mathbb R}}(X)$)
of all complex-valued
(resp., real-valued)
continuous functions on $X$,
$S\subset M$
a complex (resp., real) function space on $X$,
${\{\phi_{n}\}}$ a
sequence of unital linear contractions
from $M$ into
$C(Y)$ (resp., $C_{{\mathbb R}}(Y)$),
and $\phi_{\infty}$ a
linear isometry
from $M$ into $C(Y)$ (resp., $C_{{\mathbb R}}(Y)$).
We show,
under the assumption that ${\mit\Pi}_{N} \subset {\mit\Pi}_{T}$,
where ${\mit\Pi}_{N}$ is
the Choquet boundary for
$N=\mathop{\rm Span}
(\bigcup_{1\le n\le \infty}N_n)$,
$N_n=\phi_{n}(M)\ (n=1,2,\ldots , \infty)$,
and ${\mit\Pi}_{T}$ the Choquet boundary for
${T=\phi_{\infty}(S)}$,
that ${\{\phi_{n}(f)\}}$ converges pointwise to
$\phi_{\infty}(f)$
for any $f\in M$ provided $\{\phi_{n}(f)\}$
converges pointwise to ${\phi_{\infty}(f)}$
for any $f\in S$;
that ${\{\phi_{n}(f)\}} $
converges uniformly
on any compact subset of ${{\mit\Pi}_N} $
to
$\phi_{\infty}(f)$
for any $f\in M$ provided ${\{\phi_{n}(f)\}} $
converges uniformly to ${\phi_{\infty}(f)}$
for any $f\in S$; and that,
in the case where $S$ is a function algebra,
$\{\phi_n\}$ norm
converges to $\phi_{\infty}$
on $M$ provided ${\{\phi_{n}(f)\}}$
norm converges to $\phi_{\infty}$
on $S$.
The proofs are in the spirit
of the original one for
the theorem of Korovkin.
