Here, page $34$, the notation $l^{\infty}(\widetilde{X}) $ where $\widetilde{X} = \{ (p,q) \in X^2 : p \neq q \}$.
Question: Why the input of $f \in l^{\infty}(\widetilde{X})$ is element from $\widetilde{X}$? I thought the definition of $l^{\infty}(\widetilde{X})$ is the set $\{ x = (x_n)_{n \in \mathbb{N}} \in \prod \widetilde{X} :\}$ where the norm is $\| x \|_{\infty} = \sup_{n \in \mathbb{N}}{||(p,q)||}$.
Usually, if $I$ is any abstract set, $\ell^\infty(I)$ is the Banach space of all scalar-valued bounded functions $f$ defined on $I$.
For the definition you give, I would rather write $\ell^\infty(\mathbb N, \widetilde X)$ ("bounded $\widetilde X$-valued sequences", i.e. "bounded $\widetilde X$-functions defined on $\mathbb N$"); but perhaps this is not a standard notation. More generally, given a normed space $X$ one could consider $\ell^\infty (I, X)$ for any abstract set $I$: this would be the space of all bounded $X$-valued functions defined on $I$. With this notation, we then have $\ell^\infty(I)=\ell^\infty(I,\mathbb K)$, where $\mathbb K=\mathbb R$ or $\mathbb C$.