Background Info: I have been studying (self-study) Real Analysis for a few weeks now, and with the help of people from here I have, comfortably, made it 3/4 of the way through the first chapter on Real Numbers. However, yesterday I hit a bit of a mental block when the book started to cover Inner Product Spaces. I am starting to wonder if I should delay my studies in Real Analysis to work on Abstract Algebra, first.
My Questions:"is Abstract Algebra a necessary for a thorough understanding of Real Analysis"? Also, I have listed my math background below am I missing anything else that i might need for Real Analysis? Ultimately, should I continue my study in Real Analysis, regardless of some my short comings in math and try to pick stuff up along the way?
My Math Background: Calc 1, Calc 2, Cal 3, Linear Algebra, Discrete Mathematics (Mathematical Logic), Elementary Number Theory, Statistics, Econometrics, and all the prerequisite for the preceding classes. My favorite classes were Discrete Mathematics and Number Theory, because I enjoyed reading and writing proofs.
I did well in Linear Algebra, at least computationally well, but I never really understood the notion of a vector space. I felt that by only using $\mathbb R ^m$ spaces I couldn't get a good feel for what a vector space really is, I need to see it in a more abstract framework. Also, I have picked Geometry up along the way I have never had a formal course in Geometry.