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I'm reading Folland's Real Analysis to learn some basic functional analysis. I read through his section Normed Vector Spaces and could make my way through most of the exercises I attempted. I am reading through the next section on Linear Functionals (dual spaces, Hahn-Banach) and am finding the exercises very difficult. I had glanced through Folland's sections on measure theory and found the exercises rather easy; I though this book was at an appropriate level.

I know Folland is more of a general analysis book; I just want to learn some of the basics (the cornerstone theorems, Hilbert spaces, Lp spaces) and so I did not think it necessary to delve into a specialized book. Is there a step down from Folland that still covers this material thoroughly-but-not-too-thoroughly? Or does anyone have any recommendation for a place to find easier problems to get myself started before tackling what I think are harder ones in Folland? The books I have seen seem to be beyond my level (e.g. Conway) or more applied books that may skip out on some of the theory (e.g. Luenberger). Perhaps my problem is just that I do not have much of an intuition with this material!

Thanks.

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Did you have a look at this question? –  t.b. Oct 28 '11 at 5:03
    
Yes. Kreyszig looked promising but perhaps too geared to physicists/engineers? –  rallen Oct 28 '11 at 5:09
    
I don't know, I haven't read that book myself. It seems to cover the standard material, though. I just mentioned that thread in case you hadn't seen it before asking. Maybe this MO-thread is more helpful? I'm pretty sure that MO has more such threads, but I couldn't find them right now. –  t.b. Oct 28 '11 at 5:17
    
If you can digest the measure theory part of Folland, maybe Conway's or Lax's Functional Analysis is a good choice. –  Jonas Teuwen Oct 30 '11 at 8:41

3 Answers 3

up vote 4 down vote accepted

George F. Simmons' Introduction to Topology and Modern Analysis. Its really a good book for a beginner (who has some knowledge in Real Analysis) to understand basic Topology and Functional Analysis. The exercises in here are rather easier to solve than those in Folland or any other book. This book gives lot of motivation in the beginning of every chapter which you would definitely find useful. But you may not find $L^p$-spaces in a greater detail.

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Thanks. This looks promising. –  rallen Oct 28 '11 at 5:41
    
I second this, I find Simmons to be one of the best introductory functional books--especially if you are interested in some of the more structural results such as the Gelfan-Naimark theorem. –  Alex Youcis Oct 28 '11 at 5:50

One cannot be realistic, honest to himself, in approaching Folland's manual on functional anlysis, with merely three or four light-weight calculus courses, two college-level linear algebra course and one ordinary differential equations course for biologist. At which profession do you focus? and at which university level? Folland's book is too deep (and too long to understand) in the pure mathematics, for most of the B.Sc. degrees. His book is at intermediate graduate level, thus supposing that you already have learned thorougly in his 'Advanced Calculus' and have done all the exercises. Thus also supposing that you have read his two textbooks on Fourier series & ordinary differentiel equations, and on PDE. Folland's edited teachings are aimed at the future mechanics-applied mathematician, or the future PhD in Mathematics: Mechanical Enginnering._____________________________________________________________________________In applied mathematics, one professional speciality is Strength of Materials. In the book 'Mathematical Foundations of Elaticity' by Marsdan, you could have a look at the contents. The functional analysis is mentionned in several places, therein. Marsdan asks the reader to have learned his book on advanced calculus. I have the idea that a consequent & responsable author always publishes a revised far-advanced calculus textbook before throwing his/her readers in (infinitesimal)analysis, moreover functional analysis. Neverthless it is helpfull to put some easy readings between: Visual Approach to PDE (Springer Verlag editor), and PDE (by DuChateau; Dover publ.)

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'Mathematical Foundation of Elasticity' written by Marsdan for any person who hopes to become an expert in strength of materials, explicitly states on its back cover that it will teach the functional analysis from the start. First edited in 1983 then reprinted by Dover publication at a low price. A 1.5-inch thick, yet not isoteric textbook.

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