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Korovkin's first theorem states that if  $  ( L _ {n} ) _ {n \geq  1 }  $
is an arbitrary sequence of positive linear operators on the space  $  C [ 0,1 ] $
of real-valued continuous functions on the interval  $  [ 0,1 ] $(
cf. [[Continuous functions, space of|Continuous functions, space of]]; [[Linear operator|Linear operator]]) and if

$$ 
{\lim\limits } _ {n \rightarrow \infty } L _ {n} ( h ) = h  \textrm{ uniformly  on  }  [ 0,1 ]
$$

for all  $  h \in \{ e _ {0} ,e _ {1} ,e _ {2} \} $, 
where  $  e _ {k} ( t ) = t  ^ {k} $(
$  0 \leq  t \leq  1 $, 
$  k = 0,1,2 $), 
then

$$ 
{\lim\limits } _ {n \rightarrow \infty } L _ {n} ( f ) = f  \textrm{ uniformly  on  }  [ 0,1 ]
$$

for all  $  f \in C [ 0,1 ] $.

The statement of Korovkin's second theorem is similar to that of the first theorem, but  $  C [ 0,1 ] $
is replaced by  $  C _ {2 \pi }  ( \mathbf R ) $(
the space of  $  2 \pi $-
periodic real-valued functions on  $  \mathbf R $, 
endowed with the topology of [[Uniform convergence|uniform convergence]] on  $  \mathbf R $) 
and  $  h $
is taken from the set  $  \{ e _ {0} ,  \cos  ,  \sin  \} $.

These theorems were proved by P.P. Korovkin in 1953 ([[#References|[a3]]], [[#References|[a4]]]). In 1952, H. Bohman [[#References|[a2]]] had proved a result similar to Korovkin's first theorem but concerning sequences of positive linear operators on  $  C [ 0,1 ] $
of the form

$$ 
L ( f ) = \sum _ {i \in I } f ( a _ {i} ) \phi _ {i} ,  f \in C [ 0,1 ] ,
$$

where  $  ( a _ {i} ) _ {i \in I }  $
is a finite set of numbers in  $  [ 0,1 ] $
and  $  \phi _ {i} \in C [ 0,1 ] $(
$  i \in I $). 
Therefore Korovkin's first theorem is also known as the [[Bohman–Korovkin theorem|Bohman–Korovkin theorem]]. However, T. Popoviciu [[#References|[a5]]] had already proved the essence of the theorem in 1950.

Korovkin has tried to generalize his first theorem by replacing  $  \{ e _ {0} ,e _ {1} ,e _ {2} \} $
with other finite subsets of  $  C [ 0,1 ] $. 
He has shown that if a subset  $  \{ f _ {1} \dots f _ {n} \} \subset  C [ 0,1 ] $"
behaves like"   $  \{ e _ {0} ,e _ {1} ,e _ {2} \} $, 
then  $  n > 2 $(
Korovkin's third theorem). Moreover, he showed that a subset  $  \{ f _ {0} ,f _ {1} ,f _ {2} \} \subset  C [ 0,1 ] $"
behaves like"   $  \{ e _ {0} ,e _ {1} ,e _ {2} \} $
if and only if it is a [[Chebyshev system|Chebyshev system]] of order two.

The Korovkin theorems are simple yet powerful tools for deciding whether a given sequence of positive linear operators on  $  C [ 0,1 ] $
or  $  C _ {2 \pi }  ( \mathbf R ) $
is an approximation process. Furthermore, they have been the source of a considerable amount of research in several other fields of mathematics (cf. [[Korovkin-type approximation theory|Korovkin-type approximation theory]]).

See [[#References|[a1]]] for a modern and comprehensive exposition of these results and for (some) applications.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Altomare,   M. Campiti,   "Korovkin-type approximation theory and its application" , ''de Gruyter studies in math.'' , '''17''' , de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bohman,   "On approximation of continuous and analytic functions"  ''Ark. Math.'' , '''2'''  (1952–1954)  pp. 43–56</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.P. Korovkin,   "On convergence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk. SSSR (N.S.)'' , '''90'''  (1953)  pp. 961–964  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Korovkin,   "Linear operators and approximation theory" , Gordon&amp;Breach  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Popoviciu,   "On the proof of the Weierstrass theorem using interpolation polynomials"  ''Lucr. Ses. Gen. Stiintific.'' , '''2''' :  12  (1950)  pp. 1664–1667</TD></TR></table>