Suppose I am interested in operators $T:X\to Y$, with $X$ and $Y$ both separable Hilbert spaces. The operator norm of such $T$ can then be taken as $$ \|T\| = \sup_{\|x\|_X\leq 1}\|Tx\|_Y. $$ Since the spaces are separable, there exists a sequence $\{x_n\}$ and $\{y_m\}$ dense in the unit balls of $X$ and $Y$. It would appear then, that we can write $$ \|T\|=\sup_{n}\sup_m |(Tx_n,y_m)_Y| = \sup_{m}\sup_n |(Tx_n,y_m)_Y|. $$ In the above, I am representing the $Y$ norm as $$ \|y\|_Y = \sup_{\|\tilde y\|\leq 1}|(y,\tilde y)_Y| $$
Is there anything wrong with this intuition?
Your equation $$ \|T\|=\sup_{n}\sup_m |(Tx_n,y_m)_Y| = \sup_{m}\sup_n |(Tx_n,y_m)_Y|. $$ is indeed correct. It works because the relevant norms are continuous and because of the density of the sequences in the unit balls.