I know that to check whether a linear operator is continuous or not we have to check if the operator norm is bounded. $$T: V\to W$$,
$$\vert\vert \ T \vert\vert= \sup_{f \in V}\frac{\vert\vert \ Tf \vert\vert}{\vert\vert \ f \vert\vert} $$
There is just one thing I do not understand. I see in some solutions that they calculate the supremum over unit circle; ( $\vert\vert \ f \vert\vert=1$)
$$\vert\vert \ T \vert\vert= \sup_{\vert\vert \ f \vert\vert=1} \vert\vert \ Tf \vert\vert$$
When and why are they equivalent? Why are we allowed to evaluate the supremum only on a unit circle?
Thanks.