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A.3. The hybrid norm on ℂ [018H]

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A.3. The hybrid norm on ℂ{\mathbb{C}}

Denote by ℂhyb{\mathbb{C}}_{\mathrm{hyb}} the Banach field (ℂ,∥⋅∥hyb)({\mathbb{C}},\|\cdot\|_{\mathrm{hyb}}), where the hybrid norm is defined as

∥⋅∥hyb:=max{|⋅|0,|⋅|∞},\|\cdot\|_{\mathrm{hyb}}:=\max\{|\cdot|_{0},|\cdot|_{\infty}\},

with |⋅|0|\cdot|_{0} the trivial absolute value and |⋅|∞|\cdot|_{\infty} the usual absolute value.

The elements of the Berkovich spectrum ℳ⁡(ℂhyb){\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}}) are of the form |⋅|∞ρ|\cdot|_{\infty}^{\rho} for ρ∈[0,1]\rho\in[0,1], interpreted as the trivial absolute value |⋅|0|\cdot|_{0} for ρ=0\rho=0. This yields a homeomorphism ℳ⁡(ℂhyb)≃[0,1]{\mathcal{M}}({\mathbb{C}}_{\mathrm{hyb}})\simeq[0,1].

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