The sequence spaces c_0 and ell-infinity

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with absolute value in the real case and modulus $\mathrm{\mid }a+ib\mathrm{\mid }=\sqrt{a^2+b^2}$ in the complex case. A scalar sequence here is a function $x:\mathbb{N}\to \mathbb{K}$, including index zero. Let

${\mathrm{\ell }}^{\mathrm{\infty }}=\{x=(x_n{)}_{n\in \mathbb{N}}:{\sup }_{n\in \mathbb{N}}\mathrm{\mid }{x}_n\mathrm{\mid }<\mathrm{\infty }\},c_0=\{x\in {\mathrm{\ell }}^{\mathrm{\infty }}:x_n\to 0\},$

both equipped with $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}={\sup }_{n\in \mathbb{N}}\mathrm{\mid }{x}_n\mathrm{\mid }$. Thus $c_0$ is a specified linear subspace of the bounded-sequence space ${\mathrm{\ell }}^{\mathrm{\infty }}$.

Here $x_n\to 0$ means that for every real $\epsilon >0$ there is $N\in \mathbb{N}$ such that $\mathrm{\mid }{x}_n\mathrm{\mid }<\epsilon$ for all $n\ge N$. Addition and scalar multiplication are coordinatewise. The scalar triangle inequality makes bounded sequences and null sequences linear spaces and gives the triangle inequality for the displayed supremum norm. Absolute homogeneity follows coordinatewise, and a zero supremum forces every coordinate to vanish.