Korovkin's theorem

Let $B_{n}, n \geq 1$, be Bernstein operators:

Is it true that $B_{n} \rightrightarrows I$?

Let $\left\{ f_{n}: n \geq 1\right\}$ be a sequence of linear nonnegative functionals on a real space $C(\left[a, b\right])$ (see problem 3.42), and $f(x) = x(c), x \in C(\left[a, b\right])$, where $c \in \left[a, b\right]$ is fixed. Denote $e_{k}(t) = t^{k}, t \in [a, b], k = 0,1,2, y(t) = (t-c)^{2}, t \in [a, b]$. Prove that

A linear operator $A: C(\left[a, b\right]) \rightarrow C(\left[a, b\right])$, where $C(\left[a, b\right])$ is a real space, is called nonnegative if for all $x \in C(\left[a, b\right])$ such that $x(t) \geq 0$ for all $t \in [a, b]$, we have $(A x)(t) \geq 0$ for all $t \in [a, b]$. Prove that

Let $X = L_{p}(\left[0,1\right])$ or $X = C(\left[0,1\right])$, the operator $K_{n}: X \rightarrow X$ assigns to each function $x \in X$ the Kantorovich polynomial

where $x_{n k} = (n+1) \int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}} x(t) d t, 0 \leq k \leq n, n \geq 1$. Prove that

(P. P. Korovkin, 1959). Let $L_{n} \in \mathcal{B}(C[0,1]), n \in \mathbb {N}$, be positive operators (i.e., for any nonnegative function $x \in C[0,1]$ its images $L_{n} x$ are also nonnegative functions). Prove that if $L_{n}(1) \rightarrow 1, L_{n}(t) \rightarrow t$, and $L_{n}\left(t^{2}\right) \rightarrow t^{2}$, then $L_{n} x \rightarrow x$ for any function $x \in C[0,1]$ (i.e., $L_{n} \xrightarrow {s} I$).