I'm an economics undergrad student, and I'm self-studying mathematical analysis. While I do keep many references around, I aim to mainly work through Pugh's Real Mathematical Analysis and Baby Rudin (the latter for the exercises). Before, I worked through Tao's Analysis 1's first 5 chapters with relative ease (I've done >90% of the exercises); besides time constraints, that's why I decided to tackle the more challenging aforementioned texts.
In both books, I was able to do most problems regarding cuts and the L.U.B property, but I've struggled with some exercises related to the Euclidean Spaces' sections. In particular, the exercises where some geometric intuition seems to be useful (like showing the largest ball one can fit inside a given cube in $R^m$, and vice-versa), which I assume I'm lacking. Furthermore, problems like Rudin Ch1 Ex16 are taking me much more time than, say, proving the existence of logarithms.
I know for certain that geometry isn't required for analysis, but would it be useful for me? If so, what quick references would you suggest? Should I study linear algebra first, even if I'm not having difficulties with inner products and norms? If the answer to both is "no", should I just spend more time doing the aforementioned exercises?
EDIT: My goal is to complete the first three chapters of Pugh's text. Specifically, I want to see: a construction of the real number system, euclidean spaces, metric space theory and a bit of topology, continuity, differentiation and the Riemann integral. Time permitting I'll study linear algebra anyway and return for the multivariable part of Pugh, but at the moment I'm not worried about this.
Best Answer
If you are wanting to learn the first three chapters of Pugh's Real Mathematical Analysis, then I don't think it would be necessary to study any formal geometry for this. Certainly it helps to have some geometric intuition and to be able to visualise the problem; this is a skill you will acquire with experience. As for linear algebra, I doubt this will be necessary either.
All you really need in order to learn the fundamentals of real analysis (sequences, series, continuity, differentiability, Riemann integration) is a good grounding in the properties of the real numbers and a degree of confidence handling inequalities. With metric spaces, you'll be doing a lot of fiddling around with unions and intersections of sets using De Morgan's laws.
As for your final question, I think you'll be fine continuing with the exercises as is. Pugh's book looks to me like a friendly and self-contained introduction that would be accessible to first year mathematics undergraduates fresh from school, so should be ideal for self-study.
Good luck! :)