01 January 1970 3 7K Report

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.

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