Ask HN: If I work with applied mathematics should I still study analysis?

tl;dr yes.

A lot of applied mathematics (physics, quantitative finance, some machine learning) rely a lot on analysis. In a more general light, I would advice against skipping bricks of something that big and that close to your direct field because "it is useless".

I can give you one data point. I have worked on computer vision (implementing, training, and running convolutional neural nets), and have also applied machine learning methods on natural language processing. I have never taken analysis and I do not feel like I am missing out. In most applied machine learning fields nowadays, the "math" of the it is used more as a gateway into intuition, not really anything rigorous. For example people in machine learning research nowadays use the word "manifold" a lot, but I would wager most do not have a rigorous background in topology. In applied machine learning, the details of your problem domain matters more than understanding the derivation of your tools, this is true even in computer vision, which used to be a lot more rigorous before convolutional neural nets. However, if you are looking to take a rigorous course in probability, I would recommend analysis since it is the foundation of probability, you cannot derive a lot of basic theorems in probability without it. I took a rigorous class in probability once without analysis, and the gaps in knowledge was Clear everyday. If you are determined to tackle theoretical course just as a confidence builder, then I would recommend linear algebra since it has far wider applications. Linear algebra is also a must if you want to understand many optimization tools used in machine learning.

As for finance, I have a no direct knowledge here, but I do know when companies like D.E. Shaw recruit here at Penn for quantitative analysts, a strong math background (including analysis) is a must. Whether they use it on a day to day basis is unclear. However, taking one analysis class will not be sufficient for such firms, they are looking for many years of formal schooling and in some cases, publications.

Yea - you don't have to go immensely deep into Analysis, but having some experience in it will give you an understanding of what makes the algorithms "tick" and when they break-down.

This is especially true for things that involve numerical analysis - approximation techniques/diff eq's/etc.

And just for clarity - while any class on Analysis is mostly proofs, it's not really about the study of proofing techniques. It will use proofs as a vehicle to discuss _analytic_ properties of functions (how limits behave at infinity, etc).

If you want to go deep on stochastic processes / probability theory - useful in finance - then you'll need it for sure. Otherwise probably not. Whether "needing" to know it is a good reason for not learning it is another question. If you're asking the question then you should probably just learn at least a little for general interest.

Studying analysis gives you more tools to model changing systems. The more tools you have, the easier it is to abstract things. That's really all applied mathematics is, abstracting stuff to things you can do mathematical operations on. When counting sheep, all you're really do is abstracting the sheep as units and then performing simple arithmetic.

So why should you have all these tools to abstract stuff? Because you want to do something new and innovative. If this really isn't that interesting to you then don't bother but every time I find a new way to model something and try it out (often with unsuccessful result, though) I get giddy with joy. The few times I come up with something new and people actually value what I've contributed with it feels like heaven, however small these contributions end up being.