Why do we denote the sequence space as $l^\infty$ My question is really simple. I didn't understand the call the sequence space as $l^\infty$. I know this is from the $l^p$ spaces denotation but I didn't understand the connection between both.

$l^p$ space


  Let $p\ge 1$ a fixed real number. By definition , each element in the
   space $l^p$ is a sequence $x=(x_j)=(x_1,x_2,\ldots)$ of numbers such
   that $|x_1|^p+|x_2|^p+\ldots$ converges and the metric is defined by 
  
  $$d(x,y)=\bigg(\sum_{j=1}^{\infty}|x_j-y_j|^p\bigg)^{1/p}$$


$l^{\infty}$ space


  Let $X$ be a set of all bounded sequences of complex numbers; that is,
   every element of $X$ is a complex sequence $x=(x_1,x_2,\ldots)$ such
   for all $j=1,2\ldots$ we have $|x_j|\le c_x$, where $c_x$ is a real
   number. We choose the metric defined by
  
  $$d(x,y)=sup_{j\in \mathbb N}|x_j-y_j|$$
We denote it because $$\lim_{p\to\infty} \left(\sum_{j=1}^\infty |x_j-y_j|^p\right)^{\frac1p} = \sup_{j\in\mathbb N} |x_j-y_j|$$
We denote it because $$\lim_{p\to\infty} \left(\sum_{j=1}^\infty |x_j-y_j|^p\right)^{\frac1p} = \sup_{j\in\mathbb N} |x_j-y_j|$$