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These are extended lecture notes for a mini course at the Spring School on Non-Archimedean Geometry and Eigenvarieties held at Heidelberg University in March 2023. The goal of the course is to explain a modern take on the eigenvariety machine in the language of adic spaces. For this we build on the theory developed in the first week of the school. We also explain some basic selective non-archimedean functional analysis.
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