Is there anyone here knows what korovkin theory is all about
It is about function approximation. For instance it gives a short proof of Weierstrass's theorem that every periodic continuous function is the uniform limit of trigonometric polynomials.
be the space of continuous real functions on a compact topological space.
A minimizing subspace
means a linear subspace
containing the function 1, and such that for any point
there is a function in
having a strict minimum at
is any closed bounded real interval, and
is the quadratic polynomials.)
A positive linear operator on
is one that maps non-negative functions to non-negative functions.
Korovkin's Theorem: Let
be a sequence of positive linear operators on
, such that
uniformly for each
in a minimizing subspace. Then the same happens for each
A proof and examples are in my 1973 book "Basic methods of linear functional analysis".