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Graph of a mapping

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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)
How to Cite This Entry:
Graph of a mapping. B.A. Pasynkov, A.A. Konyushkov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Graph_of_a_mapping&oldid=18835
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098