This is a snapshot from a book on Optimal control and partial differential equations.
My doubts are:-
Is the D'($\Omega$) space metrizable with the pseudo-topology?
What is the formal definition of scalar product between spaces namely D($\Omega$) and D'($\Omega$)?
Why is the scalar product between spaces given by equation 1 like that?
Notation
$\Omega$ is an open set of n dimensional Euclidean space and $\Gamma$ is it's boundary.