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Relationship between $2 \to 2$ norm and $\infty \to 2$ norm

It is more a question on norms on spaces and not on operators as $\| \cdot \|_{N_1, N_3} \leq C \| \cdot \|_{N_2,N_3}$ if and only if $N_2 \leq \frac{1}{C} N_1$. Doing so if $l_{x,y}(z) = \left< x, z …
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