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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

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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.

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