Ich bin der Meinung, dass ... .
Ich bin der Ansicht, dass ... .
What is the case in the above examples?
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These are predicative genitives. This used to be a productive feature of several Indo-European languages and hence appears to be a very old phenomenon (Dal/Eroms 2014: 31). If you read even relatively recent literary material like Schiller plays, you will encounter this quite frequently. Note that predicative genitives could/can serve different functions. Compare:
Today, predicative genitives can be found only in certain fixed contexts, such as einer Ansicht/Meinung/Auffassung/Überzeugung/... sein (= be of an opinion/view/conviction/...); des Wahnsinns sein (= be mad); gleicher Art/gleichen Typs/... sein (= be of the same kind); guter Dinge sein (= be in good spirits); etc. You would usually learn about such phrases as part of your language studies or from a dictionary.