1 question in Theorem 10 of section Spectral Theory of Hoffman Kunze

Question

I am self studying Linear Algebra from Textbook Hoffman and Kunze and I have a question in Theorem 10 of Chapter 9 .

Adding it's image:

How it's clear that for every $\alpha$ in $E_{j}V $ f(T) $\alpha$ = f($c_{j} $) $\alpha$ ? Can anyone please tell reasoning behind it.

Answer 1

If $\alpha\in E_jV$, then $\alpha=E_jv$ for some vector $v$. Therefore $$ f(T)\alpha=\sum_if(c_i)E_iE_jv=f(c_j)E_j^2v=f(c_j)\alpha. $$