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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?

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