# Cauchy Schwarz inequality with 1 norm

Here is the argument I am making

By Holder's inequality, we have for $$\frac{1}{p} +\frac{1}{p^*} = 1$$

$$\langle A, B\rangle \leq ||A||_p||B||_{p^*}$$

The Schatten p-norms also obey $$||A||_p \geq ||A||_q$$ for any $$1\leq p \leq q\leq \infty$$ according to these notes (Section 2.3.1, bottom of page 20). Therefore, I should be able to choose $$p = 1$$ and $$p^* = \infty$$ and obtain

$$\langle A, B\rangle \leq ||A||_1||B||_{\infty} \leq ||A||_1||B||_{1}$$

Yet, this answer seems to suggest that my argument is invalid. I don't fully understand if this has to do with finite vs infinite dimensional space but can someone explain what I am missing?