Take the operator norm as an example, \begin{aligned} \|X\|_{a,b}&=\sup\{ \|Xu\|_a|\ \|u\|_b\le 1 \}\\ &=\sup\{ \|Xu\|_a|\ \|u\|_b= 1 \}\\ &=\sup \frac{\|Xu\|_a}{\|u\|_b}\\ \end{aligned}
Why do the second equality and the third equality hold? It ''looks'' obvious, but how can we prove it mathematically? You can simplify this problem by letting $a=b=2$, i.e, to explain the following, \begin{aligned} \|X\|_{2}&=\sup\{ \|Xu\|_2|\ \|u\|_2\le 1 \}\\ &=\sup\{ \|Xu\|_2|\ \|u\|_2= 1 \}\\ &=\sup \frac{\|Xu\|_2}{\|u\|_2}\\ \end{aligned}
The first line follows from the definition of operator norm. How can we show that the second and third line hold?