Korovkin’s Theorem

In mathematics when and how a given series of functions converges to a given function is a basic question, for example, the fourier series, the power series and so on, these problem can be view as a kind of approximation.

As our first choice, I would like to share the Korovkin’s Theorem.
Recall that $C([0,1])$ is the space of continuous functions on $[0,1]$, which is a linear space, thus we can define the linear mapping between $C[0,1]$ and $C[0,1]$, also called (linear) operators on $C[0,1]$. We shall call a operator $F$ on $C[0,1]$ be positive if

$$F(f)\geq0,\quad \forall f\geq0, f\in C[0,1].$$