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The problem of the geometric characterization of domains in a given [[Analytic space|analytic space]] that are Stein spaces (cf. [[Stein space|Stein space]]); it was posed by E.E. Levi [[#References|[1]]] for domains in the affine space  $  \mathbf C  ^ {n} $
in the following form. Let  $  D $
be a domain in  $  \mathbf C  ^ {n} $
each boundary point  $  \zeta $
of which has the following property: there is a neighbourhood  $  U $
of  $  \zeta $
in  $  \mathbf C  ^ {n} $
and a holomorphic function in  $  U \cap D $
that cannot be extended holomorphically to  $  \zeta $. 
Is  $  D $
a [[Domain of holomorphy|domain of holomorphy]]? This property is equivalent to any of the following assertions about the domain  $  D $: 
1) for no  $  \zeta \in \partial  D $
is there a sequence of bounded holomorphic surfaces  $  S _  \nu  $
that converges to a holomorphic surface  $  S $
with  $  \partial  S _  \nu  \rightarrow \partial  S $, 
$  \overline{S}\; _  \nu  , \partial  S \subset  D $, 
$  \zeta \in S $; 
2) the domain  $  D $
is pseudo-convex, that is,  $  -  \mathop{\rm log}  \rho ( z , \partial  D ) $, 
$  z \in D $, 
where  $  \rho $
is the Euclidean distance, is a [[Plurisubharmonic function|plurisubharmonic function]] in  $  D $; 
and 3)  $  D $
is a pseudo-convex manifold, that is, there is in  $  D $
a plurisubharmonic function that tends to  $  + \infty $
as  $  \partial  D $
is approached. The Levi problem for  $  \mathbf C  ^ {n} $
was affirmatively solved in 1953–1954 independently by K. Oka, H. Bremermann and F. Norguet, and Oka solved the problem in a more general formulation, concerned with domains spread over  $  \mathbf C  ^ {n} $(
cf. [[Covering domain|Covering domain]]) (see –[[#References|[6]]]). Oka's result has been generalized to domains spread over any Stein manifold: If such a domain  $  D $
is a pseudo-convex manifold, then  $  D $
is a Stein manifold. The Levi problem has also been affirmatively solved in a number of other cases, for example, for non-compact domains spread over the projective space  $  \mathbf C P  ^ {n} $
or over a Kähler manifold on which there exists a strictly plurisubharmonic function (see ), and for domains in a Kähler manifold with positive holomorphic bisectional curvature [[#References|[7]]]. At the same time, examples of pseudo-convex manifolds and domains are known that are not Stein manifolds and not even holomorphically convex. A necessary and sufficient condition for a complex space to be a Stein space is that it is strongly pseudo-convex (see [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]). Also, a strongly pseudo-convex domain in any complex space is holomorphically convex and is a proper modification of a Stein space (see , [[#References|[4]]] and also [[Modification|Modification]]; [[Proper morphism|Proper morphism]]).

The Levi problem can also be posed for domains  $  D $
in an infinite-dimensional complex topological vector space  $  E $. 
If  $  E $
is locally convex and  $  D $
is a domain of holomorphy, then  $  D $
is pseudo-convex, that is, in  $  D $
there is a plurisubharmonic function that tends to  $  + \infty $
as  $  \partial  D $
is approached. The converse theorem is false even in Banach spaces, but it has been proved for Banach spaces with a countable basis, as well as for a number of other classes of spaces  $  E $(
see ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.E. Levi,   "Sulle superficie dello spazio a 4 dimensione che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse"  ''Ann. Mat. Pura Appl.'' , '''18'''  (1911)  pp. 69–79</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  A.L. Onishchik,   "Stein spaces"  ''J. Soviet Math.'' , '''4''' :  5  (1975)  pp. 540–554  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''11'''  (1974)  pp. 125–151</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  A.L. Onishchik,   "Pseudoconvexity in the theory of complex spaces"  ''J. Soviet Math.'' , '''14''' :  4  (1980)  pp. 1363–1407  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15'''  (1977)  pp. 93–171</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.A. Fuks,   "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O. Suzuki,   "Pseudoconvex domains on a Kähler manifold with positive holomorphic, bisectional curvature"  ''Publ. Res. Inst. Math. Sci. Kyoto Univ.'' , '''12'''  (1976)  pp. 191–214; 439–445</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   H. Wu,   "Complex differential geometry" , Birkhäuser  (1983)</TD></TR></table>