Exercise 6.6.7 (Linear Algebra by Friedberg, Insel, Spence): Let $T$ be a normal operator on a finite-dimensional complex inner product space V. Suppose we have the spectral decomposition $\lambda_1T_1 + \lambda_2 T_2 + ... + \lambda_k T_k =T$. If $g$ is a polynomial, then $g(T) = \sum_{i=1}^k g(\lambda_i) T_i$.
I know that $T_i T_j =0$ for $ i \not= j$ $(T_i$ is the orthogonal projection of $V$ on a subspace $W_i$). I tried to prove this using induction, but continued to fail. Suppose that $g(x) = a_1 x + a_0$. Then, $g(\lambda_1) = a_1 \lambda_1 +a_0$, so the right hand side is $(a_1 \lambda_1 + a_0)T_1$. On the other hand, $g(\lambda_1 T_1) = a_1\lambda_1 T_1 + a_0 $. Is induction a correct way to solve the question? If not, I appreciate if you give some help.