Let be a bounded open set in whose boundary is at least smooth and let
(1)
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be a linear operator defined by
(2)
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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)
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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.