Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$.
I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ to $L^2(\Omega \times [0,T])$ and I denote by $\delta$ his adjoint operator that is the one satisfying
$$ \langle D(F),u\rangle_{L^2(\Omega\times[0,T])} = \langle \delta(u) , F\rangle_{L^2(\Omega)} $$
Now I consider the linear map $\phi$ that associated to each $F\in\mathbb{D}$ $\phi(F) = \mathbb{E}[\langle D(F),u\rangle_{L^2([0,T])}]$, for $u\in L^2(\Omega\times[0,T])$.
First I extend $\phi$ by uniform continuity on the whole space.
Then by the Riesz representation theorem I know that there exists a unique $Z\in L^2(\Omega)$ such that $\phi(F)=\mathbb{E}(ZF)$
I would like to prove the adjoint operator $\delta$ of the linear map $D$.
My idea is the following : define $\delta$ the adjoint operator by $\delta(u)=Z$ for each $F\in\mathbb{D}$, the map is well defined by the uniqueness of the Riesz representation theorem.
Does this seem correct please ?
Thank you a lot !