ScalingStacks

Verified tagged author-source HTML · 1904.03696v1 · cited publication edition alignment unverified.

00KT

Corollary 3.15. With the same hypothesis as above, for en∨​(x)∈(L⊗n)∨​(x)∖0e_{n}^{\vee}(x)\in(L^{\otimes n})^{\vee}(x)\setminus 0, one has

|en∨​(x)|FS​(∥⋅∥n)∨=max⁡{|en∨​(x)​(sn,j)|κ^​(x)⋅∥sn,j∥n−1}.\lvert e_{n}^{\vee}(x)\rvert_{\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})^{\vee}}=\max\Big\{\lvert e_{n}^{\vee}(x)(s_{n,j})\rvert_{\widehat{\kappa}(x)}\cdot\lVert s_{n,j}\rVert_{n}^{-1}\Big\}.
00KU

Proof. It suffices to note that en∨​(x)​(sn,j)=λje_{n}^{\vee}(x)(s_{n,j})=\lambda_{j}. ∎

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