Let 
be a bounded open set
in 
whose boundary 
is at least 
smooth and let
 |
(1)
|
be a linear operator defined by
 |
(2)
|
on the collection of all real-valued compactly-supported 
functions with domain in the topological
closure 
of 
. In functional
analysis, the trace operator is defined to be the extension
 |
(3)
|
of 
to functions whose domain is the Sobolev space 
.
Intuitively, the trace operator literally "traces" the boundary of a function 
. This piece of data
is of particular important when studying function spaces
and partial differential equations
due to the existence of various boundary-value parameters in these contexts.
