  In this article we propose a conception of mixed limits of functional spaces
as the case, when the upper limit (projective limit of inductive limits) and
the lower limit (inductive limit of projective limits) coincide as topological
spaces, which are generalization of inductive and projective limits of
functional spaces. We show a cases where these mixed limits are naturally
obtained as the limit spaces of non-commutative $L_p$-type spaces associated
with the sequence of operators. Also, we obtain results on the properties of
limit spaces, we show that limit spaces of Banach algebras are (LF)-spaces, if
they converge.
