TITLE: Does analysis do anything other than estimating things and discussing convergence?
QUESTION [7 upvotes]: Let me first of all declare that till a few months ago, Analysis was the subject I liked most, as I was pretty good at it and the idea was simple: you essentially bound things.
But then, I had these spurts of realizations: has Analysis nothing else to offer? Let's take a quick look at what analysis does.
Fourier analysis: Hell lot of PDE's, heat equations, etc. and hell lot of estimating things. The concept of generalized function I admit, is really nice, but that's essentially all to it. From the beginning to the end, estimate integrals or show certain functions belong to a class.
Complex analysis: This is a beautiful subject, although estimations crop up here as well, quite often. However, I do like things like Cauchy's Theorem, Morera's Theorem, etc. simply because they are NOT estimating things!
Functional analysis: Convergence on arbitrary spaces and some fairly complicated existential theorems. Due to lack of concrete integration, there is lack of estimating things but then, there's always the concept of showing convergence.
Analytic Number Theory: I got bored to death trying to study this. From the first page to the last (probably the book I chose wasn't friendly) I saw integrals being estimated.
I have no grievance towards analysis in particular, and as I have mentioned, I am actually good in it. I can grasp analysis concepts really well and my background is quite strong. However, after a point, you really begin to wonder whether a subject has anything else to offer other than estimating integrals/series and checking convergence. I, unfortunately, haven't been exposed to prospective fields of Analysis which go beyond these. So, at times, the journey has been immensely boring.
I would like to ask the community of mathematicians here: what is your opinion? I would love to know if there are topics in analysis beyond these estimations and computations, so if you know of them please do tell me.
Yes, something I missed is: why did I like analysis? Because it reduced a lot of computations I used to do as a high school student. Look at the Riemann-Lebesgue Lemma. Look at the power of Stone-Weierstrass. These are theorems that really boost my interest. But then, what about the rest?

REPLY [3 votes]: In a sense, analysis is the art of estimation (like algebra is an art of introducing structures), so don't expect a hardcore analysis paper to look anything but a chain of inequalities on the surface. The "let $\varepsilon>0$" opening sentence is as ubiquitous here as "Once upon a time" in the folk tales.
The point however is that, when facing an analytic problem, we seldom know the answer, so it is almost never in the form "Prove that $A\le \text{Const}B$". Typically it is in the form "Find a non-oscillatory condition equivalent to the boundedness of an operator with a sign changing kernel" or "Figure out what is the largest possible size of a set with some given property" or "Prove or disprove that we have a unique solution of something and that solution is behaving nicely", or "Does every operator 
in a Hilbert space have an invariant subspace", or.... Now, it is a pure matter of taste if you prefer questions of this type to something like "Is it true that the ring of matrices over a non-commutative nilpotent ring is always nilpotent?" or the other way around. However, keep in mind that in analysis, unlike algebra, we deal with things whose behavior cannot be grasped entirely. The task is always to discern the "main terms" in the behavior (be it an asymptotic formula or a geometric construction) and to convince ourselves that the "error terms" are, indeed, negligible. That latter step usually requires a lot of "boring" estimations. 
There are standard tricks, of course, and more often than not about 
5/6 of an exposition is just a careful routine check that nothing goes astray where it should not. However that routine check becomes possible only after you figured out where exactly to look and find out what is there at least in the first approximation. Alas, the discussion of how that was figured out is normally left out of the exposition and reduced to "Define ... by ... . We shall show that...", after which some long computation follows leaving the impression that the key to everything is in it, while more often than not it is just one of 15 possible implementations of the idea (and usually neither the slickest, nor the most transparent one). 
You say that you feel like you have a fairly good background in analysis by the moment.
I wonder if you have ever tried to solve a problem about which you wouldn't be sure which tools to apply, which way the answer goes, and whether the solution might be within your reach or not. If yes, I'm curious what was it. 
If not, you haven't really tried it yet and just judge analysis from the perspective of a spectator on a soccer game, who wonders if there is more to the game than hitting a spherical object by foot so that it would go into a rectangular frame. You cannot appreciate soccer unless you play it yourself and the same is with analysis (or algebra, or geometry, or probability, or computer science, or...) So, in that case I suggest you try now. The problem I'll offer has the classical flavor of "estimate an integral" just to leave no doubt that it is the very same "classical analysis" you got so bored with, but I'm not telling you anything about it either in terms of the degree of difficulty, or in terms of where it came from.
Let $P(x,y,z)=xyz+Q(x,y,z)$ be a polynomial of $3$ real variables of degree $5$ such that all coefficients of $Q$ are less in absolute value than some fixed positive $\varepsilon$ (which you are free to fix to be any small number you want).
Let $\alpha\in(0,1)$ be close to $1$ (say, $0.99$). Can we find a universal constant $C$ such that
$$
\int_B \frac{dx\,dy\,dz}{|P(x,y,z)|^\alpha}\le C
$$
where $B$ is the unit ball in $\mathbb R^3$ centered at the origin?
Let's see if you can figure it out.