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Famously, a problem which stood for many years concerning Banach spaces was whether or not every one admitted a Schauder basis. This was discussed by many mathematicians, including Grothendieck! Finally Per Enflo published a paper in the 1970s giving an explicit example of a space without such a basis and indeed solving a number of other problems simultaneously (the space has an operator on it with no nontrivial proper invariant subspace etc.).

Numerous places online and my lecture course reference this fact but I can't seem to find any exposition of it. It's been something like 50 years since this example was found: surely someone has written a (potentially very long) article describing this space's construction and its properties? Is there such a text?

Preferably, any reference given would focus on this space and would be self-contained bar assuming basics in functional analysis (Hahn-Banach, Open Mapping theorem etc.) which you might learn in a first course. If this is not possible, a book with a chapter or two on this topic would also be acceptable.

Isky Mathews
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  • For example the book by J. Lindenstrauss, L. Tzafriri: "Classical Banach spaces. I.", Chapter 2d. Also the book by Albiac and Kalton has a nice exposition on on Schauder bases etc, but does not provide the example by Enflo. – spin Nov 12 '22 at 02:20
  • While I haven't read the paper carefully, it's just 9 pages long, and at first sight it doesn't look particularly convoluted. Have you tried reading it? I'm curious if/where there are complications. – Martin Argerami Nov 12 '22 at 02:34
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    In the answer there is a link to Per Enflo,'s paper. He got a live goose in 1972 from Stanisław Mazur for solving the problem. The problem has been posted in Scottish Book. – Ryszard Szwarc Nov 12 '22 at 02:51

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