We denote
the dual space of
(with respect to the
topology, see the definition (
Dual space
)).
where
represents the action of
on
.

Remark

It will be clear from the proposition
(
Embedding of dual Sobolev
space
) of this section that the operation
is the extension of the scalar product
from
to
.

Proposition

(Representation of dual
Sobolev space) For an
there exist functions
such that

(Embedding of dual Sobolev space)
For a bounded domain
we
have
where the last embedding is understood as

Proof

The first embedding is trivial. The equality follows from the proposition
(
Riesz representation theorem
)
and
being a Hilbert space. For the last embedding observe
that
thus