# A question on linear integral equation about non degenerated bilinear form

Let $$X$$ be a Banach Space , $$X\subseteq H,\bar{X}=H$$,where $$H$$ is a Hilbert space $$i:=X\to H$$ defined by $$i(x)=x$$ and is continuous. Define $$\langle x,y\rangle=\langle ix,iy\rangle$$ then $$\langle X,X,(.,.) \rangle$$ to be a dual system.

Dual system definition :

from linear integral equations by rainer kress

Here I am trying to prove first Bilinear form. It's okay How to prove non-degenerate thing and dual system thing. Can someone explain this to me?

Thank you.