I will be taking Functional Analysis next semester and am looking for an appropriate book. I have taken Algebra (Aluffi "Algebra: Chapter 0"), Measure Theory (Elstrodt "Maß- und Integrationstheorie"), and Exterior Analysis (Tu "Introduction to Manifolds"; Wedhorn "Sheaves, Manifolds, and Cohomology"). Complex Analysis will be given next semester.

A book like Helemskii is almost perfect, however since exterior analysis is not a prerequisite, manifolds are not used. So Ideally I'm looking for a book.which uses manifolds and hopefully (but not necessarly) category theory.


  • $\begingroup$ Functional analysis concerns analysis and linear algebra on infinite-dimensional topological vector spaces. Manifolds are not not a usual way to go about this. Usual references for functional analysis are Rudin, Conway, or Werner. If you are a mathematical physicist Simon & Reed is perfect. Kolmogorov and Fomin is a "basic" book on the subject. $\endgroup$ – s.harp Mar 11 '18 at 18:29
  • $\begingroup$ Why aren't manifolds a useful concept when studying say, function spaces? Is the infinite dimensionality an issue? $\endgroup$ – MrHolmes Mar 12 '18 at 23:18
  • $\begingroup$ Manifolds are modelled on vector spaces. To understand manifolds you first need to understand the vector spaces on which they are modelled. Infinite dimensional manifolds must be modelled on infinite dimensional vector spaces, but a basic course in functional analysis is all about understanding these spaces. Infinite dimensional manifolds can come later, but are rarely considered because they don't have many applications. $\endgroup$ – s.harp Mar 13 '18 at 9:05
  • $\begingroup$ Another great book I came across was Francois Treves' "Topological Vector Spaces" $\endgroup$ – MrHolmes Jun 30 '18 at 22:45

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