et
and
are separable Hilbert spaces,
,
is dense in
and the natural injection operator
,
is continuous
(
topology is stronger:
convergence
implies
convergence).
We denote such
relationship
Consider the
form
Let
be the set of
such that the form
is continuous in
topology.
is dense in
with respect to
topology. By denseness of
in
the form extends to
.
Therefore, by the proposition (
Riesz
representation theorem
) for any
there is an element
such
that
The
is a linear operator
unbounded in
topology.
Furthermore,
By
,
continuous functions of
are well defined. In particular, the operator
is well defined. We
introduce
Proposition
(Interpolation inequality) Given
we
have
