We have discussed the following theorem:
>Let $X$ be a Hilbert space and $T:X\rightarrow X$ a bounded self-adjoint linear operator. Then there exists a unique map $$F:C(\sigma(T))\rightarrow B(X)$$ ($B(X)$ is the set of bounded linear operators) such that
>
> 1. for polynomials $p$, $F(p)=\sum_{n=0}^N a_n T^n$
>
> 2. $F$ is a $*$-homomorphism i.e. $F(\phi+\psi)=F(\phi)+F(\psi)$, $F(\phi\cdot \psi)=F(\phi)\cdot F(\psi)$, $F(\lambda \phi)=\lambda F(\phi)$, $F(\bar \phi)=F(\phi)^*$
>
> 3. $F$ is an isometry: $||F(\phi)||=||\phi||_\infty$