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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.

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up vote 3 down vote accepted

You are right for the second one. For the first, $\lVert X^n\rVert=1$, but $T(X^n)=\sum_{j=0}^n\binom njX^j$, hence $\lVert T(X^n)\rVert=\sum_{j=0}^n\binom nj=2^n$, proving that $T$ is not bounded for the given norm.

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