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Is there any long list of subspaces of $\ell ^\infty$ (The set of real bounded sequences equipped with the supremum norm). This would help to find a great many counterexamples. Moreover, I wonder how much interesting subspaces can it have.

boyler
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  • I do not think there is a particular answer to your question because, in papers, researchers create subspaces from the well-known spaces to fulfill their needs for which they can solve a certain problem. In my study for nonlinear PDEs, I face many new subspaces of spaces. – Mr. Proof Jan 11 '22 at 03:51
  • @AliJoohy Maybe you can share a highlight or two with us? What are cool subspaces that are good candidates for counterexamples? – Severin Schraven Jan 11 '22 at 05:21
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    There's this... – David Mitra Jan 11 '22 at 05:37

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