In the book I read recently(Eberhand Zeidler, Applied Functional Analysis: Applications to Mathematical Physics (Applied Mathematical Sciences 108)), when consider the extension of symmetric operator on Hilbert space, he goes through the following way:
- Suppose a linear, symmetric, strongly monotone operator $B: D(B) \subset X \rightarrow X$ is given. Define the energetic space $X_E$, which consists of the all the limits of what he calls "admissible sequence", i.e. for $u \in X$, $u \in X_E$ iff exist $\{u_n\} \subset D(B)$ , s.t. $u_n \rightarrow_X u$ and $\{u_n\}$ is a Cauchy sequence in $X_E$ (with regard to energetic norm).
- Showing $X_E$, equipped with energetic inner product, is a Hilbert Space.
- Define the energetic expansion $B_E$, where $B_E: X_E \rightarrow X^*_E$ is given by $(B_Eu)(v) = \left <u|v\right>_E$ ,$ \forall u,v \in X_E$. Prove $B_E$ is an expansion of $B:X \rightarrow X^*$, where $X^*$ is the duality space of $X$, and using Rietz representation theorem we can assume $X = X^*$.
- Define the Friedrichs Extension $A$ as following: $A: D(A) \subset X \rightarrow X$, $Au := B_Eu$, where $D(A) = \{u \in X_E| B_Eu \in X\}$. Prove $A:D(A) \rightarrow X$ is a self-adjoint and bijective operator.
My question is, since $X_E,X$ are all Hilbert Space, and $X_E \subset X$, using Rietz representation theorem, we have $X_E^* = X_E \subset X = X^*$, so how can the restriction in step 4 make sense? Is it because we use different inner product?