Whilst taking a class in functional analysis I couldn't help but feel Banach space theory was only ever taught as a natural stepping stone towards Hilbert space theory. As to the prominence and status of Banach spaces in mathematical research today, I have several questions:

  1. Are Banach spaces still being extensively studied today - are they a sizeable area of current interest? (If I were to go to the analysis group of a university am I likely to find any of its members working on problems exclusively within the context of / to do with Banach spaces?)

  2. What types of problems are currently being investigated in the area? Are there any open problems in the area or holes in the theory of these spaces that are already known for Hilbert spaces?

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    $\begingroup$ Yes, Banach spaces are still being extensively researched. However I'm not sure how "sizeable" is the area with respect to people, as I sometimes have trouble finding Banach space theorists in math departments. As for open problems, there is a brand new book by Guirao/Montesinos/Zizler covering this. However, it would probably help if you could be more specific than just Banach spaces, which is an enormous field. See here for more info: web0.msci.memphis.edu/~bzheng/Banach $\endgroup$ – Ben W Sep 30 '16 at 21:06
  • $\begingroup$ @Ben Wallis, your link is broken $\endgroup$ – Norbert Oct 1 '16 at 7:50
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    $\begingroup$ The link works for me. $\endgroup$ – sequence Oct 7 '17 at 0:30

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