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I have seen examples of the following:

  1. A sequence of duals extending forever$$S \subset S^* \subset S^{**} \subset S^{***} \subset ... \\ S = L^0 $$

  2. A sequence of duals flipping between 2 spaces $$S \subset S^* \subset S \subset S^{*} \subset ... \\ S = L^p $$

  3. A sequence of duals which is isometric to itself $$S = S^* \\ S = H $$

Are there sequences of duals that "return" every third space or something? Is there some "deep" algebraic fact that these are the only 3 possibilities?

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