If you're interested in theory, 1&2 are no brainers.
If not, I would take 1&3.
I am now deciding between which 2 of the following 3 courses to take
1) Optimisation Theory: Unconstrained and equality optimization models, constrained problems, optimality conditions for constrained extrema, convex sets and functions, duality in nonlinear convex programming, descent methods, conjugate direction methods and quasi-Newton methods.
Textbook: "S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004."
2) Real Analysis: This is an introductory course on the Lebesgue integration theory on real line. Topics include: measurable sets, measurable functions, Lebesgue integral, Fatou's lemma, monotone convergence theorem, Lebesgue dominated convergence theorem and differentiation.
Textbook: Real Analysis by H.L. Royden
3) Mathematical Economics (offered by Econ Department, and the grading is more lenient):
This course discusses basic tools in mathematical analysis and their applications to mathematical economics. Mathematical topics include sets, functions, sequences, continuity, open sets, closed sets, compact sets, maximum existence theorem, separating hyperplanes, and fixed points. Economic topics include preference, utility, demand, competitive equilibrium, and Pareto optimality
Textbook: Lecture notes by professor
I have taken Calculus I to III, Linear Algebra, ODE, Probability, Basic Real Analysis, Stochastic Processes, Stochastic Calculus. I am interested in doing macro/finance stuffs (probably theory). I am quite interested in (2) as I quite enjoy Basic Real Analysis and Measure seems necessary for theory. (3) is in fact a good choice since it covers lots of topics including optimisation and fixed points, but it may seem less sophisticated. That's why I am struggling between these courses, particularly (1) and (3).
Does anybody have some thoughts on this. Thanks a lot!!!!!!
I don't think I would put (1) before (2) necessarily - I just didn't read your post carefully. If you're interested in applied micro, I don't think there's any reason to take measure theory, while (1) will probably help you more with your first year coursework. But it seems like you are interested in finance/theory, in which case measure theory is probably necessary.
Optimization will be much more useful to help prepare you for graduate micro and the MWG material. My school had an optimization course in the econ department that was specifically for this purpose and it covered similar topics.
The title of (2) is Real Analysis but as pulsars says, it's more measure theory than analysis. It reminds me of the second course in the analysis sequence at my undergrad school. Not super useful other than for signalling that you like tough math and can do well in it.
The third option seems like stuff you would already know based on your "Basic Real Analysis" course.
Do you need to take 2? Are you sure you'll do well if you take 2? If you're certain of getting an A in all 3, then take (1) & (2) in my opinion. If it'll be harder to juggle, I'd take (1) and find a more rigorous course than (3) that interests you.
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