Definition of obtaining a copy of $l^{\infty}(2)$

Question

Currently I am reading the book 'Isometries on Banach Spaces: Vector-Valued function spaces and operator spaces.'

In chapter $7$ page $16,$ the author stated that:

$Y$ does not contain a copy of real $l^{\infty}(2)$ if and only if at least one of the norms $\| x \pm y \| <2$ whenever $\| x \| = \| y \|=1.$

Question: What is the definition of not containing a copy of $l^{\infty}(2)?$ Also, what is $l^{\infty}(2)?$

I search it on the internet but fail to obtain its definition.

Answer 1

It means that it does not contain a subspace isomorphic to $l^\infty (2)$

Answer 2

$l^\infty_2$, $l^\infty(2)$ (or $\ell_\infty^2$ in certain circles including mine) is $\mathbb{R}^2$ (or $\mathbb{C}^2$ if you prefer complex scalars) endowed with the maximum norm. More generally, $\ell_p^n$ stands for $\mathbb{R}^n$ with the $p$-norm. Curiously, the spaces $\ell_\infty^2$ and $\ell_1^2$ are isometric. This is no longer the case in higher dimensions.

By saying that a Banach space $X$ does not contain $\ell_p^n$ people usually mean that $\ell_p^n$ is not isometric to a subspace of $X$.