I am supposed to find the whether the following maps are continuous or not , if continuous then to find the $||T||$
$P$ is a vector space of polynomials . Define norm on the polynomials $p\in P$ as $\sum_{n=0}^n|a_n|$ where $a_n$ are the coefficients of the real polynomials.
I have the map $Tp(t)=p(t+1)$ and $Tp(t) =\int_0^tp(s)ds$
My attempt has been as follows for the second one :
$Tp(t)=\int_0^1p(s)ds =\sum_{k=0}^na_k\frac{t^{k+1}}{k+1}$ $\implies$ $||T(p)||=\sum|\frac{a_k}{k+1}| \le1. \sum |a_k|=||p||$
Hence its continuous because every bounded operator is continuous. and also i find $||T||=1$
For the first one after expanding the terms i get something like this term $||p(t+1)||= \sum_{k=1}^{n}\binom{n}{n-1}a_n+\sum_{k=2}^{n}\binom{n}{n-2}......+\sum_{k=n}^{n}\binom{n}{0}$
it looks like its bounded but i can't proceed from here . And i don't know if i am right ? Can anyone help me please. Thank you very much in advance.