Graph of a mapping
of a set
into a set
The subset of the product
consisting of the points
,
. If
and
are topological spaces,
is a continuous mapping and
is the projection of the topological product
onto the factor
, then the mapping
is a homeomorphism of the subspace
onto
. If
is a Hausdorff space, then the set
is closed in the product
.
B.A. Pasynkov
In the case of a real-valued function of
real arguments
and domain of definition
, its graph is the set of all ordered pairs
, where
is any point of
; in other words, it is the set of all points
in
. Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points
,
can be represented by points of the plane or space. For real-valued functions
in one real variable which have derivatives
,
, in more or less complicated examples the graph can be sketched by studying the signs of
and
. The sign of
is an indicator of the monotony of
, while the sign of
indicates the direction of convexity of the graph of the function. To obtain an idea on the graph of a real-valued function
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
; the projection of this section on the
-plane is said to be a level set of
. Similarly, for a function
defined in a domain
, the level set of
at level
, where
is an arbitrary number, is the set of all solutions of the equation
. The solutions
must be found in
. 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 and
are Fréchet spaces (cf. Fréchet space) and
is a linear mapping with a closed graph, then
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) |
Graph of a mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Graph_of_a_mapping&oldid=47129