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$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:

I know that in $L^2(0,1)$ we have that relation:

$$||T||\leq \left ( \int_0^1\int_0^1 |(5s^2t^2+2)|^2dtds\right ) ^{\frac{1}{2}}=\sqrt{\frac{50}{6}}.$$

I also have that $||T||=\sup_{||f||_2\leq1} ||Tf(s)||$ so if I take $f=1$ then this gives equality above and I am done, is that correct?

Thanks for any help