I have the following question of the proof of 9.30(c) in 'Linear Algebra Done Right' by Sheldon Axler.
Theorem 9.30 states the following: 'Suppose V is an inner product space, $T\in\mathcal{L}(V)$ is normal, and U is a subspace of V that is invariant under T. Then
(a) $U^\perp$ is invariant under T
(b) U is invariant under $T^*$
(c) $(T|_{U})^* = (T^*)|_U$
(d) I'll skip it.
When proving (a) he gets to the following matrix of T (image below). $e_1,...,e_m$ is an orthonormal basis of U, and $e_1,...,e_m,f_1,...,f_n$ is an orthonormal basis of V. A is a mxm matrix and C is an nxn matrix.
His proof of (c) is pretty straightforward, but i was wondering if one could proof (c) by saying that $M(T|_{U})^* = M(T^*)|_U$ implies $(T|_{U})^* = (T^*)|_U$?
I'll appreciate any guidence.
