Timeline for Korovkin subset of $C(\mathbb{T})$

6 events

when toggle format	what		by	license	comment
May 20, 2019 at 17:32	history	edited	YCor	CC BY-SA 4.0	fixed English and latexified, fixed typo
May 20, 2019 at 16:49	comment	added	Jochen Glueck		Note that, since all $T_n$ are positive, they map real-valued functions to real-valued functions. This implies that $\overline{T_nf} = T_n\overline{f}$ for each $n$ and each $f$. Hence, the answer is "yes", as follows from the link provided by @YemonChoi (the corresponding result is called "Korovkin's second theorem" there). By the way, this even shows that $\{1, z\}$ is a Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$.
May 20, 2019 at 16:35	history	edited	Tanmoy Paul	CC BY-SA 4.0	added 12 characters in body
May 20, 2019 at 16:32	comment	added	Tanmoy Paul		Yes, I mean, whether $\\|T_n(f)-f\\|_\infty\to 0$ for all $f\in A$ implies $\\|T_n(g)-g\\|_\infty\to 0$?
May 20, 2019 at 16:28	comment	added	Yemon Choi		Working backwards from the statement of Korovkin's theorem encyclopediaofmath.org/index.php/Korovkin_theorems I assume that Tanmoy's use of "uniformly" should be "uniformly on $K$". My own preference would be to say $\Vert T_n(f)-f\Vert_\infty \to 0\forall\,f\in A \implies \Vert T_n(g)-g\Vert_\infty\to 0 \forall\,g\in C(K)$.
May 20, 2019 at 16:10	history	asked	Tanmoy Paul	CC BY-SA 4.0