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An [[Analytic space|analytic space]] the local rings of all points of which are normal, that is, are integrally-closed integral domains. A point  $  x $
of an analytic space  $  X $
is said to be normal (one also says that  $  X $
is normal at  $  x $) 
if the local ring  $  {\mathcal O} _ {X,x} $
is normal. In a neighbourhood of such a point the space has a reduced and irreducible model. Every simple (non-singular) point is normal. The simplest example of a normal analytic space is an [[Analytic manifold|analytic manifold]].

In what follows the (complete non-discretely normed) ground field  $  k $
is assumed to be algebraically closed. In this case the most complete results on normal analytic spaces have been obtained (see [[#References|[1]]]) and a normalization theory has been constructed [[#References|[2]]] that gives a natural link between arbitrary reduced analytic spaces and normal analytic spaces. Let  $  N ( X) $
be the set of points of an analytic space  $  X $
that are not normal and let  $  S ( X) $
be the set of singular points of  $  X $(
cf. [[Singular point|Singular point]]). Then:

1)  $  N ( X) $
and  $  S ( X) $
are closed analytic subspaces of  $  X $, 
and  $  N ( X) \subset  S ( X) $;

2) for  $  x \in X \setminus  N ( X) $,

$$ 
 \mathop{\rm dim} _ {x}  S ( X)  \leq    \mathop{\rm dim} _ {x}  X - 2
$$

(that is, a normal analytic space is smooth in codimension 1);

3) if  $  X $
is a complete intersection at  $  x $
and if the above inequality holds, then  $  X $
is normal at that point.

A normalization of a reduced analytic space  $  X $
is a pair  $  ( \widetilde{X}  , v) $, 
where  $  \widetilde{X}  $
is a normal analytic space and  $  v:  \widetilde{X}  \rightarrow X $
is a finite surjective analytic mapping inducing an isomorphism of the open sets

$$ 
\widetilde{X}  \setminus  v  ^ {-} 1 ( N ( X))  \rightarrow  X \setminus  N ( X).
$$

The normalization is uniquely determined up to an isomorphism, that is, if  $  ( \widetilde{X}  _ {1} , v _ {1} ) $
and  $  ( \widetilde{X}  _ {2} , v _ {2} ) $
are two normalizations,

$$ 

\begin{array}{rcr}
\widetilde{X}  _ {1}  &  \mathop \rightarrow \limits ^  \phi     &\widetilde{X}  _ {2}  \\
{} _ {v _ {1}  }  \searrow  &{}  &\swarrow _ {v _ {2}  }  \\
{}  & X  &{}  \\
\end{array}

$$

then there exists a unique analytic isomorphism  $  \phi :  \widetilde{X}  _ {1} \rightarrow \widetilde{X}  _ {2} $
such that the diagram commutes. The normalization exists and has the following properties. For every point  $  x \in X $
the set of irreducible components of  $  X $
at  $  x $
is in one-to-one correspondence with  $  v  ^ {-} 1 ( x) $. 
The fibre at  $  x \in X $
of the direct image  $  v _ {*} ( {\mathcal O} _ {\widetilde{X}  }  ) $
of the structure sheaf  $  {\mathcal O} _ {\widetilde{X}  }  $
is naturally isomorphic to the integral closure of the ring  $  {\mathcal O} _ {X,x} $
in its complete ring of fractions.

The concept of a normal analytic space over  $  \mathbf C $
can be introduced in terms of analytic continuation of holomorphic functions [[#References|[3]]]. Namely, a reduced complex space is normal if and only if Riemann's first theorem on the removal of singularities holds for it: If  $  U \subset  X $
is an open subset and  $  A \subset  U $
is a closed analytic subset not containing irreducible components of  $  U $, 
then any function that is holomorphic on  $  U \setminus  A $
and locally bounded on  $  U $
has a unique analytic continuation to a holomorphic function on  $  U $. 
For normal complex spaces Riemann's second theorem on the removal of singularities also holds: If  $   \mathop{\rm codim} _ {x}  A \geq  2 $
at every point  $  x \in A $, 
then the analytic continuation in question is possible without the requirement that the function is bounded. A reduced complex space  $  X $
is normal if and only if for every open set  $  U \subset  X $
the restriction mapping of holomorphic functions

$$ 
\Gamma ( U, {\mathcal O} _ {X} )  \rightarrow \ 
\Gamma ( U \setminus  S ( X), {\mathcal O} _ {X} )
$$

is bijective. The property of being normal can also be phrased in the language of local cohomology — it is equivalent to  $  H _ {S ( X) }   ^ {1} {\mathcal O} _ {X} = 0 $(
see [[#References|[5]]]). For any reduced complex space  $  X $
one can define the sheaf  $   {\mathcal O}  tilde _ {X} $
of rings of germs of weakly holomorphic functions, that is, functions satisfying the conditions of Riemann's first theorem. It turns out that the ring  $   {\mathcal O}  tilde _ {X,x} $
is finite as an  $  {\mathcal O} _ {X,x} $-
module and equal to the integral closure of  $  {\mathcal O} _ {X,x} $
in its complete ring of fractions. In other words,  $   {\mathcal O}  tilde _ {X} = v _ {*} ( {\mathcal O} _ {\widetilde{X}  }  ) $, 
where  $  v:  \widetilde{X}  \rightarrow X $
is the normalization mapping.

A normal complex space can also be characterized in the following manner: A complex space is normal if and only if every point of it has a neighbourhood that admits an analytic covering onto a domain of  $  \mathbf C  ^ {n} $(
see [[#References|[3]]], [[#References|[8]]]).

A reduced complex space  $  X $
is a [[Stein space|Stein space]] if and only if its normalization  $  \widetilde{X}  $
has this property (see [[#References|[4]]]). To normal complex spaces one can extend the concept of a Hodge metric (see [[Kähler metric|Kähler metric]]). Kodaira's projective imbedding theorem [[#References|[6]]] carries over to compact normal spaces with such a metric.

In algebraic geometry one examines analogues of normal analytic spaces: normal algebraic varieties (see [[Normal scheme|Normal scheme]]). For algebraic varieties over a complete non-discretely normed field the two concepts are the same (see [[#References|[7]]], [[#References|[1]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Houzel, "Géometrie analytique locale I" , ''Sem. H. Cartan Ann. 13 1960/61'' , '''2''' (1963) pp. Exp. 18–21 {{MR|}} {{ZBL|0121.15906}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Komplexe Räume" ''Math. Ann.'' , '''136''' (1958) pp. 245–318 {{MR|0103285}} {{ZBL|0087.29003}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Narasimhan, "A note on Stein spaces and their normalisations" ''Ann. Scuola Norm. Sup. Pisa'' , '''16''' (1962) pp. 327–333 {{MR|0153870}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Y.T. Siu, G. Trautmann, "Gap sheaves and extensions of coherent analytic subsheaves" , Springer (1971) {{MR|0287033}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" ''Math. Ann.'' , '''146''' (1962) pp. 331–368 {{MR|}} {{ZBL|0178.42702}} {{ZBL|0173.33004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1960) {{MR|0120249}} {{ZBL|0121.27801}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> B.A. Fuks, "Theory of analytic functions of several complex variables" , '''1''' , Amer. Math. Soc. (1963) (Translated from Russian) {{MR|0174786}} {{MR|0168793}} {{ZBL|0138.30902}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8 {{MR|0387634}} {{ZBL|0265.32008}} </TD></TR></table>