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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449701.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449702.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449703.png" />''
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The subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449704.png" /> of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449705.png" /> consisting of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449707.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g0449709.png" /> are topological spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497010.png" /> is a continuous mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497011.png" /> is the projection of the topological product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497012.png" /> onto the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497013.png" />, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497014.png" /> is a homeomorphism of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497015.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497017.png" /> is a [[Hausdorff space|Hausdorff space]], then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497018.png" /> is closed in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497019.png" />.
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'' $  f:  X \rightarrow Y $
 +
of a set  $  X $
 +
into a set  $  Y $''
 +
 
 +
The subset $  \Gamma $
 +
of the product $  X \times Y $
 +
consisting of the points $  ( x, f ( x)) $,  
 +
$  x \in X $.  
 +
If $  X $
 +
and $  Y $
 +
are topological spaces, $  f $
 +
is a continuous mapping and $  p: X \times Y \rightarrow X $
 +
is the projection of the topological product $  X \times Y $
 +
onto the factor $  X $,  
 +
then the mapping $  p $
 +
is a homeomorphism of the subspace $  \Gamma $
 +
onto $  X $.  
 +
If $  Y $
 +
is a [[Hausdorff space|Hausdorff space]], then the set $  \Gamma $
 +
is closed in the product $  X \times Y $.
  
 
''B.A. Pasynkov''
 
''B.A. Pasynkov''
  
In the case of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497021.png" /> real arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497022.png" /> and domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497023.png" />, its graph is the set of all ordered pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497025.png" /> is any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497026.png" />; in other words, it is the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497028.png" />. Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497030.png" /> can be represented by points of the plane or space. For real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497031.png" /> in one real variable which have derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497033.png" />, in more or less complicated examples the graph can be sketched by studying the signs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497035.png" />. The sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497036.png" /> is an indicator of the monotony of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497037.png" />, while the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497038.png" /> indicates the direction of [[Convexity|convexity]] of the graph of the function. To obtain an idea on the graph of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497039.png" /> in two real variables, the method of sections may be employed: One studies the sections of the graph by certain planes, in particular by planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497040.png" />; the projection of this section on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497041.png" />-plane is said to be a level set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497042.png" />. Similarly, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497043.png" /> defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497044.png" />, the level set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497045.png" /> at level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497047.png" /> is an arbitrary number, is the set of all solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497048.png" />. The solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497049.png" /> must be found in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497050.png" />. A level set may prove to be empty. If a level set is a line or a surface, it is known as a level line or a level surface of the function.
+
In the case of a real-valued function $  f $
 +
of $  n $
 +
real arguments $  x _ {1} \dots x _ {n} $
 +
and domain of definition $  E  ^ {n} $,  
 +
its graph is the set of all ordered pairs $  (( x _ {1} \dots x _ {n} ), f ( x _ {1} \dots x _ {n} )) $,  
 +
where $  ( x _ {1} \dots x _ {n} ) $
 +
is any point of $  E  ^ {n} $;  
 +
in other words, it is the set of all points $  ( x _ {1} \dots x _ {n} , f ( x _ {1} \dots x _ {n} )) $
 +
in $  E  ^ {n} \times \mathbf R $.  
 +
Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points $  ( x, f ( x)) $,
 +
$  ( x, y, f ( x, y)) $
 +
can be represented by points of the plane or space. For real-valued functions $  f $
 +
in one real variable which have derivatives $  f ^ { \prime } $,  
 +
$  f ^ { \prime\prime } $,  
 +
in more or less complicated examples the graph can be sketched by studying the signs of $  f ^ { \prime } $
 +
and $  f ^ { \prime\prime } $.  
 +
The sign of $  f ^ { \prime } $
 +
is an indicator of the monotony of $  f $,  
 +
while the sign of $  f ^ { \prime\prime } $
 +
indicates the direction of [[Convexity|convexity]] of the graph of the function. To obtain an idea on the graph of a real-valued function $  z $
 +
in two real variables, the method of sections may be employed: One studies the sections of the graph by certain planes, in particular by planes $  z = c $;  
 +
the projection of this section on the $  xy $-
 +
plane is said to be a level set of $  z $.  
 +
Similarly, for a function $  f $
 +
defined in a domain $  E  ^ {n} $,  
 +
the level set of $  f $
 +
at level $  c $,  
 +
where $  c $
 +
is an arbitrary number, is the set of all solutions of the equation $  c = f ( x _ {1} \dots x _ {n} ) $.  
 +
The solutions $  ( x _ {1} \dots x _ {n} ) $
 +
must be found in $  E  ^ {n} $.  
 +
A level set may prove to be empty. If a level set is a line or a surface, it is known as a level line or a level surface of the function.
  
 
''A.A. Konyushkov''
 
''A.A. Konyushkov''
  
 
====Comments====
 
====Comments====
An extremely important theorem in functional analysis is the so-called [[Closed-graph theorem|closed-graph theorem]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497052.png" /> are Fréchet spaces (cf. [[Fréchet space|Fréchet space]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497053.png" /> is a linear mapping with a closed graph, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044970/g04497054.png" /> is continuous. Many generalizations of this result are known (see [[#References|[a1]]]).
+
An extremely important theorem in functional analysis is the so-called [[Closed-graph theorem|closed-graph theorem]]: If $  X $
 +
and $  Y $
 +
are Fréchet spaces (cf. [[Fréchet space|Fréchet space]]) and $  f : X \rightarrow Y $
 +
is a linear mapping with a closed graph, then $  f $
 +
is continuous. Many generalizations of this result are known (see [[#References|[a1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. De Wilde,  "Closed graph theorems and webbed spaces" , Pitman  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. De Wilde,  "Closed graph theorems and webbed spaces" , Pitman  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR></table>

Latest revision as of 19:42, 5 June 2020


$ f: X \rightarrow Y $ of a set $ X $ into a set $ Y $

The subset $ \Gamma $ of the product $ X \times Y $ consisting of the points $ ( x, f ( x)) $, $ x \in X $. If $ X $ and $ Y $ are topological spaces, $ f $ is a continuous mapping and $ p: X \times Y \rightarrow X $ is the projection of the topological product $ X \times Y $ onto the factor $ X $, then the mapping $ p $ is a homeomorphism of the subspace $ \Gamma $ onto $ X $. If $ Y $ is a Hausdorff space, then the set $ \Gamma $ is closed in the product $ X \times Y $.

B.A. Pasynkov

In the case of a real-valued function $ f $ of $ n $ real arguments $ x _ {1} \dots x _ {n} $ and domain of definition $ E ^ {n} $, its graph is the set of all ordered pairs $ (( x _ {1} \dots x _ {n} ), f ( x _ {1} \dots x _ {n} )) $, where $ ( x _ {1} \dots x _ {n} ) $ is any point of $ E ^ {n} $; in other words, it is the set of all points $ ( x _ {1} \dots x _ {n} , f ( x _ {1} \dots x _ {n} )) $ in $ E ^ {n} \times \mathbf R $. Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points $ ( x, f ( x)) $, $ ( x, y, f ( x, y)) $ can be represented by points of the plane or space. For real-valued functions $ f $ in one real variable which have derivatives $ f ^ { \prime } $, $ f ^ { \prime\prime } $, in more or less complicated examples the graph can be sketched by studying the signs of $ f ^ { \prime } $ and $ f ^ { \prime\prime } $. The sign of $ f ^ { \prime } $ is an indicator of the monotony of $ f $, while the sign of $ f ^ { \prime\prime } $ indicates the direction of convexity of the graph of the function. To obtain an idea on the graph of a real-valued function $ z $ in two real variables, the method of sections may be employed: One studies the sections of the graph by certain planes, in particular by planes $ z = c $; the projection of this section on the $ xy $- plane is said to be a level set of $ z $. Similarly, for a function $ f $ defined in a domain $ E ^ {n} $, the level set of $ f $ at level $ c $, where $ c $ is an arbitrary number, is the set of all solutions of the equation $ c = f ( x _ {1} \dots x _ {n} ) $. The solutions $ ( x _ {1} \dots x _ {n} ) $ must be found in $ E ^ {n} $. A level set may prove to be empty. If a level set is a line or a surface, it is known as a level line or a level surface of the function.

A.A. Konyushkov

Comments

An extremely important theorem in functional analysis is the so-called closed-graph theorem: If $ X $ and $ Y $ are Fréchet spaces (cf. Fréchet space) and $ f : X \rightarrow Y $ is a linear mapping with a closed graph, then $ f $ is continuous. Many generalizations of this result are known (see [a1]).

References

[a1] M. De Wilde, "Closed graph theorems and webbed spaces" , Pitman (1978)
[a2] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
How to Cite This Entry:
Graph of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Graph_of_a_mapping&oldid=47129
This article was adapted from an original article by B.A. Pasynkov, A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article