I'm studying solution given by transposition and i would like to clear somethings.
Assume we have the following, let $T < \infty$ and $Q=\Omega \times] 0, T[$, let $\varphi$ given in a Hilbert space $H_1(Q)$ and assume $v \in H_2(Q)$ is solution of the following adjoint problem \begin{equation} \begin{array}{c} A^{*} v-v^{\prime}=\varphi \text { en } Q , \\ C_{j} v=0 \text { on } \Sigma,\\ v(x, T)=0, \quad x \in \Omega . \end{array} \end{equation}
Define the space $X$ as follows \begin{equation*} X(Q)=\left\{v \in H_2\mid v \,\, \text{satisfies the above system} \right\}. \end{equation*}
So under certain hypothesis on operators $A$ and $C_j$ we have the following result
Theorem: The operator $P^{*}\left(=A^{*}-\frac{\partial}{\partial t}\right)$ is an isomorphism of $X(Q)$ in $H_1(Q)$.
So the idea is now to Transpose the isomorphism to get the following
Solution given by transposition: Let $v \rightarrow L(v)$ a continuos anti lineal form on $X(Q)$, then there exist a unique $u \in H_1^{\prime}$ such that $\left\langle u, A^{*} v-v^{\prime}\right\rangle =L(v) \quad \forall v \in X(Q)$. Where $\langle,\rangle$ denothe the bracket duality between $H_1$ and $(H_1)^{\prime}$
Such $u$ it's the solution to the original system.
With this in mind, my questions are the following
I would like to know what does mean to "transpose" a linear map, in this case to transpose given isomorphism.
Is there any way to know why such form on $X(Q)$ satisfy $\left\langle u, A^{*} v-v^{\prime}\right\rangle =L(v) \quad \forall v \in X(Q)$, I mean I'm not asking for a proof, just for an idea of why or how we got such equality.
Hopefully u can help me and the given information is enough. I wrote everything in terms of a parabolyc system but the idea of transposition works for any kind of systems.