How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces?
Theorem 1. Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon > 0$ be given. There is at least one matrix norm $||\cdot||$ such that $\rho(M) \leq ||M|| \leq \rho(M) + \epsilon$ where $\rho(M) = \max\{|\lambda_1(M)|, \dots , |\lambda_n(M)|\}$ denotes the spectral radius of $M$.
Theorem 2. If $P \in \mathbb{C}_{n\times n}$ and $S\in \mathbb{C}_{n\times n}$ are such that $P = P^2$ and $PS = SP$ then $\rho(PS) \leq \rho(S).$
I have used these results in finite dimensional spaces and want to use them in a Banach space.