How do I show that $\ell^{\infty}$ is a normed linear space?

Question

How do I show that $\ell^{ \infty}$ is a normed linear space, where $\ell^{ \infty}$ is define as $$\|\{a_n\}_{n=1}^{\infty}\|_{\ell^\infty}=\sup_{1 \leq k \leq \infty} |a_k|?$$ There are three properties that I need to check in order for this to be a normed linear space. Nonnegativity, positive homogeneity, and the triangle inequality.

I am having trouble working with the supremum. Any idea will be greatly appreciated thanks

Answer 1

For the triangle inequality, suppose that $\{a_n\},\{b_n\}\in \ell^{\infty}$. For each index $n$ we have $$|a_n+b_n|\leq |a_n|+|b_n|\leq \sup_{m}|a_m|+\sup_m|b_m|=\|a\|_{\infty}+\|b\|_{\infty}.$$ Then taking the supremum over all $n$ shows that $$ \|a+b\|_{\infty}\leq \|a\|_{\infty}+\|b\|_{\infty}. $$

Answer 2

Let $(a_n),(b_n) \in \ell^\infty$ Then we have for each $k$:

$$|a_k+b_k| \le |a_k|+|b_k| \le ||(a_n)||_\infty+||(b_n)||_\infty.$$

This shows that $(a_n+b_n) \in \ell^\infty$ and that

$$||(a_n+b_n)||_\infty \le ||(a_n)||_\infty+||(b_n)||_\infty.$$

Answer 3

Well begin by noticing that for $a,b\in \mathbb{R}$ and $(x_n),(y_n)\in \ell^\infty$ you have that $(ax_n+by_n)$ is bounded, therefore it belongs in $\ell^\infty$ as well.

For $\|\cdot\|_{\infty}$ to be a norm, check that it is non-negative and $\|x_n\|_{\infty}$ becomes zero only when $(x_n)=0$.

$\|\cdot\|_{\infty}$ also satisfies $\|ax_n\|_{\infty}=|a|\|x_n\|_{\infty}$ and use the triangle inequality for the absolute value to show that $\|x_n+y_n\|_{\infty}\leq \|x_n\|_{\infty}+\|y_n\|_{\infty}$. These prove $\|\cdot\|_{\infty}$ is a norm on $\ell^\infty$.