Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?

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    $\begingroup$ Your question is very general. What kind of answer do you expect? What is the context? $\endgroup$ – Fredrik Meyer May 6 '11 at 22:07
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    $\begingroup$ First you should start learning what a vector space is, then what a Hilbert Space is, then what a Banach space is and then what a Sobolev space is. Don't try to learn what a Sobolev space is when you don't know what a vector space is. You would fail. $\endgroup$ – Jonas Teuwen May 6 '11 at 22:09
  • $\begingroup$ To add to Jonas' plan: while vector spaces, Hilbert spaces, and Banach spaces can be learned completely abstractly (I wouldn't recommend it, but they can), Sobolev spaces by definition requires familiarity with Lebesgue integration and other aspects of real analysis. $\endgroup$ – Willie Wong May 7 '11 at 2:08

Clearly, vector spaces are the most general notion among these, because they can be defined over any field and all, Banach, Hilbert and Sobolev spaces are vector spaces.

Banach spaces make sense over any normed field (e.g. $\mathbb{R}$ or $\mathbb{C}$). I've only encountered Hilbert spaces over the reals or the complex numbers so far. Every Hilbert space is a Banach space and every Banach space is a normed vector space.

Sobolev spaces are quite a bit more special. They are a class of function spaces, usually defined for open subsets of $\mathbb{R}^n$ or sometimes on manifolds. Sobolev spaces are usually Banach spaces, but they can be Hilbert spaces as well (for the exponent $p = 2$).

Finally, you omitted the important classes of locally convex spaces and Fréchet spaces among many other things. Except for general vector spaces all the mentioned classes are (Hausdorff) topological vector spaces, but I don't think it's worthwhile to indulge in name-dropping. Without further information about your goal, it's hard to tell what kind of answer you are looking for. I'd recommend having a look at any introductory text on functional analysis where most of these classes of spaces are touched upon.

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    $\begingroup$ People use $p$-adic Banach spaces. (I'm not sure exactly what for, though.) $\endgroup$ – Qiaochu Yuan May 6 '11 at 22:40
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    $\begingroup$ @Qiaochu: They arise naturally for instance in the investigation of Fourier analysis on $p$-adic groups (geometric, representation theoretic and number theoretic questions with $p$-adics). The fields $\mathbb{Q}_{p}$ or their completions $\mathbb{C}_p$ are a normed fields, that's what I meant when saying e.g. $\mathbb{R}$ or $\mathbb{C}$, but I already mentioned too many words without explanation... An introductory book on that is Schneider's $p$-adic functional analysis. $\endgroup$ – t.b. May 6 '11 at 22:43
  • $\begingroup$ What book on Functional Analysis would you recommend? $\endgroup$ – dzhelil May 7 '11 at 21:57
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    $\begingroup$ @Dzhelil: It depends a lot on your background and your interests. I find Conway's book quite good and my personal favorite is Pedersen's Analysis now. A short and very good introduction is given in Zimmer: Essential results of functional analysis. $\endgroup$ – t.b. May 7 '11 at 22:10
  • $\begingroup$ Well, I have learned the subject from Conway but I had a hard time understanding it but maybe you're smarter than I am. I like Lax's book. $\endgroup$ – Jonas Teuwen May 12 '11 at 13:42

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