Is there an inner product over $\Bbb C^2$ such that the linear transformation $T: \Bbb C^2 \to \Bbb C^2$ defined by the matrix $ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $ is normal in respect to it?
I tried to use the spectral thorem, and I found out that $T$ is not diagonalizable. Does that mean there is no such inner product?
Same goes over $\Bbb R$, and the linear transformation $T:\Bbb R^2 \to \Bbb R^2$ defined by $ \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $ . In this case I found that there are no real eigen values, so same as before, is it true that there is no such inner product?
No, there is no such inner product. If there was, it would follow from the spectral theorem that $T$ would be diagonalizable. This is about your first $T$.