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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
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1 Answer

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
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