How to determine if I'm talented enough to study math? After getting Bachelor's degree from Computer Science I changed my field to Applied Mathematics. The previous degree was mostly programming-oriented, so I know quite a lot about software engineering, databases, Linux etc., but not so much about maths.
To be honest, we had just basic single-variable calculus, basics of linear algebra and a little of graph theory.
After finishing my B.S. I got some awards for my bachelor thesis and because of that I was offered a job in our research centre. I changed my field to mathematics to be more useful there (most of my colleagues are concentrating on computational mathematics).
The problem is, that while I'm studying for final exams of this term, I recognized, that I don't understand things thoroughly and some of them at all. The biggest problem for me is Functional Analysis, where I'm simply stuck on very basic concepts and I have to look up something for almost every lemma or proof I want to understand.
I takes tremendous amount of time and even then I feel like I know nothing about it, because in every subject there are exercises I don't understand and proofs I'm not able to invent on my own.
The truth is, I was able to pass somehow all the tests so far and to finish all the projects with full score.
But still, I feel I'm not very confident in this and so I thought about the possibility I'm simply not talented enough.
So, is there any way to find out if my problem is caused by the lack of talent or just by gaps in my knowledge I'll be able to fill one day?
I'm mean something like what amount of knowledge one should be able to grasp in one half of a year, one year etc.

I've read several questions about studying maths here on SE. For example:
Grasping mathematics
How to effectively study math?
Steps to Re-Learn Mathematic the right way
 A: Stop doubting yourself right now:  you're talented enough.  The only question is will you work hard enough?  Listen to Terence Tao, the idea of lacking "talent" should be done away with:  https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
You've got to find the problem areas where you lack understanding and spend some time carefully learning everything.  Analysis and analysis proof-writing is difficult and requires time.  Sit down with a copy of Spivak's Calculus or Rudin or something and go over every part with a fine-toothed comb until you understand everything.  There's no other way to do it.  If you have the willpower to make yourself concentrate, and can come away with some understanding, you shouldn't have any problems.
Also, talk to people about math.  Go to your math department (or a message board or something) and have a discussion about what you want to know.  That's the easiest way to learn, from other mathematicians.
But, most importantly, STOP DOUBTING YOURSELF!
A: Ok so 
1) Dont let anybody else (exams, professors, university committiees) decide whether you are talented enough.
2)Forget the question of whether you are talented, and ask yourself if it interests you. 
3) And most importantly are you willing to put in the time and effort to overcome the obstacles.
A: Before beginning my mathematical studies at the university I had the same question and asked my former maths teacher if he recommended me to study maths. His answer was simply: "If you can imagine yourself working for about 80 hours per week on mathematical problems, then do it, otherwise let it be!" I thought he wanted to tell me jokes, since he knew that I was somewhat lazy...So I decided to ask a professor at the university where I was going to study maths, what I could expect. He did not know about my laziness, but told me almost the same (about 70 hours per week). Then I knew what I had to expect...
Now, as a mathematician, I know what the two people wanted to say: First of all, studying maths consists of hard work, talent can be helpful but if you really enjoy solving tricky problems and puzzling, developing strategies and so on, then I guess you will also have enough talent...
Another fact which occured to me: The more you already know (in a certain field) the more you become aware of the vast amount of things you do not know yet!
It is no reason for fear if you come across some open questions which you have not yet an answer for, but can be merely a sign for developing consciousness for the subject. 
When I was in this situation I found it very helpful to take a "bird's eye view" of the stuff, to find the "links" between the theorems and lemmas, to find out what proposition is needed for the proof of theorem xyz and so on, not to loose myself in the details. It is helpful to have a "red line" along which the different topics can be arranged in an appropriate way in order to get a closer understanding of the results and their relationship one to another.
