This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as:
$$(x,y)\mapsto xy$$

(Let indicate that map with $B$), we have:

$$\|B(x,y)-B(a,b) \|=\|B(x-a,b)+B(a,y-b)+B(x-a,y-b)\| \\ \leq\|B\|\|x-a\|\|b\|+\|B\|\|a\|\|y-b\|+ \|B\|\|x-a\|\|y-b\|$$

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> $$\leq\|B\|(2\|(a,b)\|\|(x-a,y-b)\|+\|(x-a,y-b)\|^2)=\|B\|\|(x,y)-(a,b)\|(2\|(a,b)\|+ \|(x,y)-(a,b)\|)$$
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Where $\|B\|$ signifies the norm of product and $\|\ x \|=\| \ x \|\_{\infty}= \max\_{1 \leq i \leq k} \|x^i\|\_i$, product product of norms on $X^1 \times X^2$.

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> We take a $\delta$ such that at the same time $0< \delta <1 $ and $(2\|(a,b)\|+1)\delta < \epsilon $ are true. From there: $$\|(x,y)-(a,b)\|< \delta \implies \|B(x,y)-B(a,b)\|< \epsilon.$$
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