How to prove each natural projection for a basis has norm at least 1 ? Also, i want to prove
"Let $(x_n)$ be a basis for a Banach space $B,$ define a new norm for the space by $\||x\|| = \sup_n \|P_n x\|,$ and let B' be the same vector space with the new norm. Then $(x_n)$ is a monotone basis for $B'$ i.e. i want to prove that $\sup_n \||P_n\||=1.$"
I move as following: $$\||{P_n x}\|| = \displaystyle \sup_m \|P_m P_n x\| = \displaystyle \sup \{P_1 x, P_2 x, ..., P_n x\}= \displaystyle \sup_{1 \leq m \leq n} \|P_m x\| \leq \||{x}\||.$$ So, $\sup_n \||P_n\|| \leq 1$. I have the same problem how to prove $\sup_n\||P_n\|| \geq 1.$
Note that the two norms are equivalent $\|.\| \leq \||.\|| \leq \sup_n \|P_n\| \|.\|.$