How to be confident that my background for my research is adequate?
To expand a bit on the comments saying this is a normal feeling (which it is): learning something in a class is well and good, but many things will not truly click until you have to use them. To take a random example, I learned and understood the proof of the Arzela-Ascoli Theorem in my third year of undergrad, but I thought it was just a neat result. I certainly forgot the statement and proof multiple times, and had to look it up. It wasn't until halfway through grad school that I realized this theorem (and/or the idea behind it) underlies pretty much every compactness theorem for function spaces, making it (in my judgment) one of the most important theorems in analysis. After that, it was not so hard to remember.
Due to this to me it feels I should read a textbook for 2 weeks then start doing the research.
Others have suggested to only read a book when you need a particular lemma/proof/etc., however I am conflicted on whether this is good advice.
Some subjects you'll need to know in detail, others you can pick up on the fly. With experience, you'll get better at distinguishing one from the other, but for now your advisor can give you guidance about this.
If your primary symptom of concern is that you lack an encyclopaedic knowledge of your subject matter then I wouldn't worry - it would take a rare specialist with a very special brain to have that level of knowledge of an academic subject. Checking definitions is common in research and it is rare to memorise academic material in a level of detail that would render checks unnecessary. For most academics it is usual --over time-- to develop a solid knowledge of key concepts and results, with the ability to check more tangential matters, or particulars of definitions and proof, by reference to academic texts and papers.
The best thing you can do is to prioritise the importance of concepts and results in your field, and try to commit important core concepts and results to memory as well as possible, while making note of where you can get information you are likely to forget. Unless you are going to become a specialist mathematician it is generally unnecessary to memorise proofs of theorems, and even for people in this category, they might know a few proofs off by heart, but for most they will remember that a certain technique is used, without remembering all the details. For mathematical work it is useful to remember the substance of important theorems and remember the techniques that are applied to prove them. It is perfectly okay to forget the exact conditions of theorems or the exact details of proofs, but if they are really important, you should try to remember the substance of them, and have the capacity to find the details when needed.
Over time you will find that you learn your field more comprehensively and you become more used to the methods and techniques used in that field. For a mathematical subject you will usually find that most things are proved with variations of a relatively small kit of proof techniques, and you will start to recognise them well enough that you can remember roughly how to prove things without much cognitive load. As a result of this you will find that you can absorb material faster and so it will become much quicker to refresh forgotten knowledge with textbooks and papers. A refresher that took you two weeks at the end of your PhD might take only a single morning once you are an experienced researcher. That takes time!
This is definitely a normal feeling. To address the feeling of not being prepared, you might want to realistically revisit where your skills and comprehension were a few years ago. If you can feel the progress you have made already, you may be more likely to believe that you will make the progress from the confusion you feel right now to having defended a dissertation.
Do you ever have a chance to tutor undergraduates in upper-level classes? Could you sit in on an instructor's office hours for classes you took at the end of undergrad? This might help you see that you have made a lot of progress.
It may feel like your colleagues are ahead of you. You might have a few of them who truly are, who know their own specialty and everyone else's specialties at an expert level. If you look at the rest of your cohort who are not doing Computational Algebra, though, you probably know more about the details of your field than they know about it.
You can remind yourself of what you do know. Map out some of the big concepts that are related to the work you want to do. Actually draw a picture about how ideas link up. This can be half brainstorming, and half a chance to see that you have a much broader knowledge of your area, and how things fit together, than you used to. (You can come back and do this at the end of every term, perhaps, and you'll see how your picture changes and grows.)
You've chosen to surround yourself with smart, talented peers and professors. Comparing yourself to them may make you feel like you're behind, but comparing yourself to yourself will remind you of your ability to learn and grow.