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most recent 30 from mathoverflow.net 2018-06-24T03:46:36Z

Constructive proof of existence of non-separable normed space

Fri, 03/09/2018 - 05:48

I am looking for a constructive proof of one of the following two statements. If they are not constructively provable, I would be very thankful for an explanation as to why that is so (i.e., at which point in a proof must non-constructive means be employed?).

1. There exists a normed space X such that for all Y $\subset$ X, if Y is denumerable, then Y is not dense in X.

2. There exists a normed space X such that for all Y $\subset$ X, if Y is dense in X, then Y is not denumerable.

I'd consider a proof constructive if it includes no applications of the: