what is expected from a PhD student? As a PhD student in applied mathematics or mathematics in general, are you expected to be able to prove every problem, for example, in an elementary real analysis book? I know it sounds silly but I am wondering if I have high expectations of myself...
For example, some theorems in an elementary real analysis book have proofs which are lengthy and time consuming to understand. Still I try to understand them and manage to do so. But if you ask me about the theorem, say after 2 months, there is a high chance I'd have forgotten how to prove it. I might have ideas but I cannot solve it in totality.
What I'm asking is, I guess, is it normal to understand something in math and forget about it? Or does the fact that you forgot about it/how to do it an indication of not having understood the subject in full in the first place?
 A: Absolutely! When I look back at some of my old papers, I find it hard to follow what's going on. 
The point about research is that you're pushing yourself and mathematics to its limits. It's impossible to maintain such a high standard over such a bredth of knowledge.
PhD research requires independent and original thought. No-one cares if you can solve all of the problems in someone else's book. What good is it to you anyway? 
Insight, vision, imagination, understanding  and passion are what are needed to be a successful researcher. 
It's not about being a memory man who can memorise the 1,001 tricks required to solve various problems. As long as you understand the material when it's infront of you, you'll be fine. You're not expected to remember millions of pages of trivia; that's what libraries and journals are for!
A: Practically nobody can work all the problems, unless they have lots of time and perhaps some help.  You want to do enough proofs and computations so that the subject sinks in for you.  "Enough" depends both on you and the subject.
Remembering proofs is not really where you are headed.  What you want is to know how to tackle a new subject and make progress.  One thing the proofs you work on show you is a lot of techniques for doing that, and you can put those in your mathematical toolkit without necessarily remembering the whole proof of anything.  These proofs also show you connections, so that two ideas that seem disparate can be shown to be related.  The more connections you have in your head, the more ideas you will have about making progress with new problems, and the better your instincts will be about what direction to take.
You are learning how to think about mathematics, not memorizing a bunch of stuff you may never use.  
A: When I read a math book, and arrive at a proof I go through the following process:


*

*Try and prove the theorem without looking at the proof. Most of the time I fail.

*Read the proof and write it out by hand. The purpose of this step is to learn techniques which might be used later. 

*Summarize the proof in one or two sentences. 


The summary is what you need to remember because more often than not you can use it to reconstruct the original proof. So to answer your question, yes it is very normal to forget the details in a proof but you should understand the idea of the proof.
A: Expectations depend on the school and program. For example, some schools have qualifying exams that expect you to be able to do some of these harder real analysis proofs on the spot, other schools have no qualifying exams. Qualifying and oral exams can also be track specific. For example, if you are in the probability & statistics track of an applied math program you may be expected to understand a good amount of measure theory and functional analysis. If you are in an operations research track you may not be expected to understand a good deal of analysis.
A: An outstanding question-and I'd assume in general, you're expected to be able to teach every undergraduate course in the average mathematics program by the time you're ready to submit your thesis for review.But of course,this differs from program to program-research originality will be more important then overall general knowledge at the more prestigious universities,I'd imagine. As far as forgetting, I can't imagine even the very best mathematicians being able to remember,except vaguely,a piece of theory they knew intimately in thier graduate school days,but hadn't looked at or thought deeply about for 30 years.  
A: Have it studied and then forgotten is not a big deal.
However, you need to remember where the proof was located. This is the idea of search and re-search.
A: As you learn more things and get a wider field of experience, you start to be able to remember the ideas of proofs, and trust that you could fill in the details if need be. If you don't remember a proof, the goal is to be able to ask yourself, "why would it be true?" and then set about filling in the details of the argument. 
So the key is to read a lot of different things, different texts with different emphases, and find the proofs and explanations that are the easiest for you to understand and remember. For every single book I have, even the very best, there are proofs in it where I think, "I would rather do this a different way." 
A close reading of a particular text can be good as a primer, to boost your mathematical understanding, but I think proficiency in a subject requires you to learn in a "discourse" with several authors. 
Collect perspectives!
A: Discovering himself proofs of almost every theorem in a textbook is the heart of learning process in maths, and this is almost easy, and most enjoying part of activity for an average but motivated student. Doing a lot of problems without mastering theorem's proof is a big lie. Discovering various proofs gives true insight into the subject. The level of difficulty one has to surmount in discovering a particular theorem gives the true value to a theorem.      
A: I would say one has to work out things by hand him/herself. Commiting to memory takes years of practice. If you 'discover' a proof yourself, it gives you both the confidence and the tools to solve a new/forgotten problem. Here a saying in martial arts applies: 
From 1 thing, learn 10000 things
A: It seems to me that the expectation is not that you be able to prove all the theorems, but rather, that you be able to be able to prove all the theorems.
What I mean is, you shouldn't need to be intimately familiar with the proof of every theorem you encounter. Rather, if you were handed a theorem and proof, you should be able to read and reconstruct the proof after a relatively short period of study. 
The important quality to strive for is not detail recall. It's instead gathering enough context to quickly grasp the ideas in a proof you don't recall and then reconstruct the details from those ideas.
At least, that's my sense of what's generally expected of me as a graduate student. 
