In $\displaystyle P_{2}( \mathbb {R} ) $, let T be defined by T(f) = f', where $\displaystyle <f,g> = \int _{0}^{1} f(t)g(t)dt $

Is T normal, self-adjoint, or neither?

In $\displaystyle M_{2x2} ( \mathbb {R} ) $, define T by $\displaystyle T(A) = A^t$

Same problem here.

I don't know how to find T* from here. Please help, thanks.