I'm really blanking out (a lot of late nights these past 10 weeks).
The point of the exercise I'm about to type up is to show that the adjoint structure may possibly change when the inner product structure changes.
We define the inner product as $\langle (v_1, v_2), (w_1, w_2)\rangle = (v_1, v_2) \begin{pmatrix} 1& \frac{1}{2} \\ \frac{1}{2} & 1 \end{pmatrix} (w_1,w_2)^t$ with $(v_1,v_2)(w_1, w_2) \in \mathbb{R}^2$.
We choose vectors $v = (1,0)$ and $w = (1,0)$ and $A= \begin{pmatrix} 1&1\\0&1 \end{pmatrix}$.
Then, we can show that $⟨Av,w⟩ = ⟨(1,0),(1,0)⟩ = (1,0)(1,1/2)^t = 1$, while $⟨v,A^\dagger w⟩ = ⟨(1,0),(1,1)⟩ =(1,0)(\frac{3}{2},\frac{3}{2})^t=\frac{3}{2}$
This shows that the usual property of adjoints in the standard inner product spaces $⟨Av, w⟩ = ⟨v, A^\dagger w⟩$ does not hold.
Now, my question is how to find the adjoint of this inner product space constructed with this "unusual" inner product.
Is there a general method to finding adjoints?
Thank you!