# Operator norm and Hilbert Schmidt norm

I'm looking for a proof of

$$||T||\leq ||T||_{HS},$$ for which it is sufficient to show $$||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0$$ can someone help?

Let $(e_i)_{i\in I}$ be an an orthonormal basis, and $x =\sum_{i\in I} y_i e_i$ with norm $1$, i.e. $\sum_{i\in I} |y_i|^2 = 1$, then
$$\|Tx\| \leq \sum_{i\in I}|y_i|\|Te_i\| \leq \sqrt{\sum_{i\in I} |y_i|^2}\sqrt{\sum_{i\in I}\|Te_i\|^2} = \|T\|_{HS}$$