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There are many instances where definitions of functions appear self-referential, but are in fact consistent and well-defined. Here are a couple cases:

  • The proof of the existence of a maximal analytic continuation for a holomorphic function germ at a point on a Riemann surface: without going into details, the definition $f(\eta ) = \eta (p(\eta ))$ is used, where $p$ is the projection from a component of the disjoint union of the stalks of the sheaf of holomorphic functions on the Riemann surface, and $\eta $ is a function germ at the point $p(\eta )$.
  • In identifying a locally convex topological vector space $X$ with the space of continuous linear functionals on ${X^*}$-with-the-weak*-topology, one sends an element of $X$ to the evaluation functional, so $x(\Lambda ) = \Lambda (x)$.

Has there been a systematic study of such cases?

Also, I would appreciate seeing other examples, if you have one handy.

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Consider the $Y$ combinator of computer science, which leads me to suspect you might find some results in and/or cited by the general realm of programming language semantics. – dfeuer Jul 24 '13 at 5:45

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