In the standard proof of Hilbert projection theorem the axiom of countable choice (denoted by CC) is used. I wonder whether there is a model of ZF+ the negation of CC in which Hilbert projection theorem for Hilbert spaces that are not finitely dimensional fails. Perhaps, there are experts in both functional analysis and set theory who can answer my question easily. I would be grateful for their hint.
This is a good question.
A good place to start in answering this question is Theorem 5.1, page 28 in
FANO CONGRUENCES OF INDEX 3 AND ALTERNATING 3-FORMS
https://arxiv.org/pdf/1606.04715.pdf
More to the point, see Theorem 2.5, page 7 in
Regularization by Discretization in Banach Spaces
https://arxiv.org/pdf/1506.05425.pdf
There is close interplay between the foundational theorems of functional analysis and some form of the axiom of choice or it's equivalences, e.g. Zorn's Lemma. This led to a "constructive" school in the 1970's to establish the results of functional analysis, Hahn Banach, Open Mapping, etc. without use of the axiom of choice in any form.
I was discussing this topic the other day and when I went poking around to see the status of this approach I found the following reference which might bear on your question.
http://math.stackexchange.com/questions/783760/hilbert-projection-theorem-without-countable-choice
Dear Professors James Peters and Truman Prevatt,
I thank you for your answers and the links very much. I do not want to replace metric completeness by another kind of completeness: for instance, by: for each sequence $(F_n)$ of nonempty closed subsets of $X$ such tat $F_{n+1}\subseteq F_n$, the intersection $\bigcap_{n} F_n$ is nonempty if the sequence of diameters of F-n converges to zero. I also do not want to replace the usual notion of a closed set by another notion. To understand Hilbert's projection theorem well, it is necessary to consider standard metric completeness David Fremlin recommended to me to change the notion of completeness to get a version of Hilbert's projection theorem in ZF, To replace metric completeness by more "suitable" kind of completeness is not a good idea because it does not lead to a satisfactory, complete solution of the problem I have asked about and it does not lead to a deep understanding. It is hardly a piece of advice of kind: if this is too difficult to answer the question, just avoid difficulties by changing assumptions and leave the question without any complete answer to it. I am changing all my lectures on topology in ZFC to lectures on topology in ZF. I suspect that I will have a lot of work with such changes till the end of my life. I find it extremely important to do it well.
Once again, thank you for your kind responses very much. Eliza Wajch
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