The trace operator assigns boundary values to elements of a Sobolev space. Let 
and let 
be a bounded open set
in 
whose boundary 
is of class 
. 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 the Sobolev
space 
. The operator
extension is bounded and unique.
Intuitively, the trace operator literally records the boundary values of a function 
. This piece of data is of particular
importance when studying function spaces and partial differential equations due to
the existence of various boundary-value parameters in these contexts.
