How important is it for a physicist have an understanding of analysis?
It might be useful to know what kind of physics you're referring to.
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For the sake of discussion, suppose my plan was to study cosmology or another very math-y field.
Well, as an example, quantum mechanics makes liberal use of Hilbert spaces, so strong backgrounds in linear algebra and analysis would be very useful there. It's hard to generalize (because it depends to strongly on what, specifically, you're studying), but having, at least, an introductory real analysis course under your belt would be very useful. At the very least, it would help you understand a lot of the other mathematics you'll be using.
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