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I did a course in introductory functional analysis and liked it. Now I want to learn more functional analysis with the goal of maybe eventually doing research (phd). I tried to find out what (advanced) functional analysis is about here but the list isn't so helpful. From it I gather among other things I want to learn about spectral theory and operator theory but items like "Banach–Mazur theorem" and so on doesn't seem to be a "topic".

Where can I look to get an overview of topics in functional analysis? I want to use the information you can give me to pick a few topics and then books about them to prepare myself. Thank you.

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  • It depends on what you want to study. A great book for general (more advanced) functional analysis is Rudin's Book "Functional Analysis". It covers all the main topics (for me) to be studied, including things such as distributions (which are very important if one wants to research in PDE's), banach algebras and operator theory.

    For the latter, I recommend Sunder's and Murphy's books, but they're much more specific.

     – Luiz Cordeiro Apr 19 '13 at 15:37

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