Infinite series of sequences

Question

Let S be the set of sequences whose series converge absolutely. We define 2 norms on S: $$\| \{ a_n \}_{n=0}^{ \infty } \|_1 = \sum_{n=0}^\infty | a_n |$$ and, $$\| \{ a_n \}_{n=0}^\infty \|_{\sup} = \sup \{ |a_n|_{n=0}^\infty \} $$ Note: S is the set of sequences such that $\| a \|_1 < \infty.$ (The sup-norm is sometimes called the infinity-norm.)

Define a linear operator $\Sigma : S \to \mathbb{R}$ by: $$\Sigma \big( \{ a_n \}_{n=0}^\infty \big) = \sum_{n=0}^\infty a_n$$ Question 1: find the operator norm of $\Sigma$ using $\| . \|_1$.

Question 2: show that the operator norm of $\Sigma$ using $\| .\|_{\sup}$ is unbounded.

Can anyone help me answer this question or give me hints. Also sorry about my latex code I am new at it and very bad but I tried my best!

Answer 1

HINTS:

$1)$ Prove an easy inequality of that norm and then find an example where the equality holds.

$2)$ Think harmonically :)

Answer 2

$|\Sigma (a_n)=|\sum a_n|\leq \sum |a_n|$ so $\|\Sigma\| \leq 1$. Since $\Sigma (1,0,0,...)=1$ and $\|(1,0,0...0\|_1=1$ we see that $\|\Sigma\| \geq 1$. Hence $\|\Sigma\| = 1$.

Now $|\Sigma (1,\frac1 2 ,..\frac 1 N,0,0...)| \to \infty$ but $\| (1,\frac1 2 ,..\frac 1 N,0,0...)\|_{sup} \leq 1$ for all $N$ so $\Sigma$ is unbounded for the second norm.