# Operator Norm Question

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.