# Duality Pairing and self-adjoint operators notation

Let $X$ be a Hilbert space and let $X'$ be the dual space of $X$ with respect to the duality pairing $\langle\cdot,\cdot\rangle$.

Let $A: X \mapsto X'$ be a bounded linear operator. We assume that $A$ is self-adjoint, i.e., we have $\langle Au,v\rangle = \langle u,Av\rangle$ for all $u,v \in X$

I am confused about the meaning of being self-adjoint, since if $\langle Au,v\rangle := (Au)(v)$ then what does $\langle u,Av\rangle$ mean?

## 1 Answer

In a Hilbert space, you have the Riesz Representation Theorem, which tells you that given any $$f\in X'$$, there exists $$y\in X$$ such $$\tag{1}f(y)=\langle y,x\rangle,\ \ \ x\in X.$$ And this assignment is isometric so $$X'$$ is isomorphic to $$X$$. In practice, one thinks that $$X'=X$$ via the duality $$(1)$$.

In summary, in the case of a Hilbert space $$X$$ the duality $$\langle\cdot,\cdot\rangle$$ is precisely the inner product of $$X$$.

• Just to clarify, are we allowed to treat bilinear maps $X \times X' \mapsto R$ as inner products $X \times \ X \mapsto R$ due to $X'$ being isomorphic to $X$ ? What if Bilinear map is not Positive-definite?
– Sam
Commented Jul 14, 2016 at 15:42
• Yes (blinear if $X$ is real, sesquilinear if $X$ is complex). If $\varphi:X\times X'\to\mathbb R$ is bilinear and bounded, then for each $f\in X'$ you have $$x\longmapsto\varphi(x,f)$$ is in $X'$, so there exists $y\in X$ such that $$\varphi(x,f)=\langle x,y\rangle$$ for all $x\in X$. Commented Jul 14, 2016 at 16:27