(Sobolev space) For a subset
,
we define
where the
is the multi-index defined in the section
(
Function spaces section
) and the
derivative is understood in the weak sense, see the definition
(
Weak derivative
).

The norm for
is defined
by

The space
is the closure of
with respect to
.

We introduce the
notation

Proposition

The space
equipped with the operation
is a Banach space.