KKecon

If CV is a requirement to do Topology then I will do CV with Topology, Topology can come useful to do theory. If CV is not a requirement then I go for AMM. Honestly, I have yet to see how content in CV is useful to economics (complex numbers maybe but not properties of complex analytical function which CV focus on).

Getting an average grade will hurt you but won't, by it alone, throw your application away.

Based on my first proof-based linear algebra course (covered what you mentioned above and more i.e spectral theory, used "Linear Algebra by K. Hoffman and R. Kunze" ), it won't be as difficult to start off with no proof background in Linear Algebra than Analysis. Undergraduate Linear Algebra proofs were usually far less involved than Analysis proofs. And you will most likely be doing proofs involved in topics that you are familiar with in linear algebra than compared to analysis. My advices, if you take the honour version, are: 1) sit down and spend sometime to understand each steps of your lecture proofs and see whether you can use them to answer different questions, 2) try to come up with atleast two methods to prove each questions in your problem set, 3) always try to find an easier way to complete your proofs and 4) find counter-examples to theorems when certain conditions don't hold (this teaches you to keep sharp and not make silly mistakes due to making unintended assumptions). Keep practicing and you should get a hang on with proof techniques.

I agree with gradfuture. If RA 2 has Lebesgue Intergration then it should be highest on the list of maths courses which you wish to take to remedy your B. Take LA 2 if it covers more advanced stuff like basic functional analysis: look at the syllabus whether it mentions things like Linear Functionals or Spectral Theory. Take Complex Analysis only if you love to have new tools to solve mathematical problems (highly unlikely to be related to Economics) and already have taken RA2 and LA 2.

Won't knowledge of complex numbers covered in Cal3 and such be sufficient? From my complex analysis course, the focus was on complex analytic functions' properties which had little to no application to real variables (crazy stuff you can never justify in real variables holds for complex analytic functions). So do you need knowledge of complex analytic function to understand these topics?

Is this suppose to answer my question? (Sorry if I am being presumptuous)

Conditional on that you do well then the answer is always obvious. If you ace courses then you can qualify to apply MPhil. Will your other two math classes be proof-based? Honestly, it is more informative to assess based on what you have than what you think you will achieve, especially if you have no previous proof-based math courses experience and you are planning to do them in the future.

Honestly, stuff that are taught in applied differential equation can be easily picked up by yourself if you have some understanding of matrix algebra and some elementary analysis. If I had to choose between the two and want a higher educational return then I would go for the theoretical version.

I am curious to how complex analysis would help you in Economics. Complex differentiation and many of its property are completely different and can't be applied to real.

No one will be able to give you a solid answer to whether one can squeeze these extra curricula in. It all depends on one's ability to comprehend the material. If you want to have a 'taste' of harshness then I recommend taking courses like a high introductory level course in analysis (without taking further courses teaching you to do proofs i.e. straight out of Cal 3 or 4) or an advanced course in game theory. These courses will force you to think rigoriously and on your feet, few key ingredients to the 'harshness' in grad school.

But isn't learning by doing the main purpose of grad school?

If you are planning to do a PhD straight out of your master programs then I would recommend you not to apply to any one year UK master programs (i.e UCL and Warwick). With one year UK master programs, you won't have any grades available until the very end of the program.

Just out of interest, does functional analysis serve to provide intuition to solve Macro problems or simply technical tools to prove theorems? And usually, do they touch on functional analysis beyond basic stuff like compact operators and few key results in Hilbert and Banach Space? Thanks a bunch!

I know someone with bare minimum maths background (i.e. nothing beyond cal 3) who got offers for the program. According to him, they value your interest and knowledge of development issues more than technical skills.

When you meant Business Discipline, I thought you are talking about non-Econ PhD (since you said Econ PhD>Business Disciplines) background hence excluding business economists. My original question was why business professors are more likely to know about the applicant's skill set better than a mathematic professor. Accounting professor may have more exposure to economics but the professor is most likely restricted to examining student's skill in accounting in his/her respective course than economics. Similar idea applies to other business disciplines (excluding those involved with business economics). Surely a math professor saying "he's impressed with how the applicant is skilled in proofs" gives a better signal than what an accounting professor may likely say when the student has only taken accounting courses with the professor. From what I have seen in this board and from friends, applicants usually choose their math professors when there is a clear signal that he/she will write more than "He did well in real analysis class". Math Professors can attest your ability to problem solve difficult maths problems etc. which can signal the applicant can frame interesting questions into models and solve them.