Topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable?

Question

Could someone list the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that a new math graduate student should be familiar with? Also could someone list the recommended/standard study materials for these topics? Thanks in advance!

Edit1: Precisely I wish to ask "the topics of analysis beyond in $\mathbb{R}$ and beyond of functions of single real variable that new math graduate students are expected to be already familiar with". Thanks for the comment for clarification by Mike Haskel.

Edit2: Sorry. This question is a cross-posting. The same question is asked in Reddit r/math here: https://redd.it/3ppxn9

Answer 1

I'm going to assume you're asking what a new student might be expected to know when they arrive at graduate school. A good start would be familiarity with the following topics, to the point where the student can do proof-based exercises without assistance.

Walter Rudin gives a good treatment of real analysis at this level in his Principles of Mathematical Analysis (a.k.a. "Baby Rudin," ISBN 978-0070542358).

Metric Spaces

• Convergence, Cauchy sequences, and completeness.
• Continuity vs. uniform continuity vs. Lipschitz continuity.
• Pointwise vs. uniform convergence.

Differentiation

• Derivatives of functions $\mathbb{R^n} \to \mathbb{R^m}$ as linear maps.
• Multivariable Taylor approximation.
• Inverse and implicit function theorems.

Integration

• Formal development of Riemann integration.

Topology of $\mathbb{R}$

• $\mathbb{R}$ is the unique realization of a certain list of axioms.
• Bolzano-Weierstrass theorem (closed and bounded iff sequentially compact).
• Heine-Borel theorem (closed and bounded iff topologically compact).
• Topological connectedness.