CMI, Jan-Apr 2025, M. Sundari
Royden & Fitzpatrick (2014), Ghorpade & Limaye (2018), Apostol (1985), Rudin (1976)
Lecture notes
Normed linear spaces
- LEC CAL1 1 ✅
- The p-norm
- LEC CAL1 2 ✅
- Equivalence of p-norms
- LEC CAL1 3 ✅
- Equivalence of all norms on a finite dimensional normed linear space
- LEC CAL1 4 ✅
- Example of complete normed linear space, inner product spaces (mostly stuff covered in ALG1)
- LEC CAL1 5 ✅
- Examples of finding limits in
Metric space topology
See Royden & Fitzpatrick (2014)
- LEC CAL1 6 ✅
- Cantor intersection theorem, constructing the completion of a metric space
- LEC CAL1 7 ✅
- Finite intersection property, total boundedness, characterization of compact sets
- LEC CAL1 8 ✅
- Extreme value theorem, Lebesgue covering theorem
- LEC CAL1 9
- Separable metric spaces, Lindelöf covering theorem, Baire category theorem
- LEC CAL1 10
- Banach contraction principle
Multivariable functions
See Ghorpade & Limaye (2018) (please don’t)
- LEC CAL1 11 ✅
- Order properties of , Intervals, disks, and bounded sets
- LEC CAL1 12
- Monotonicity and Bimonotonicity, bounded variation and bivariation
- LEC CAL1 13 ✅
- Convexity and Concavity, paths, local extrema and saddle points
- LEC CAL1 14
- Intermediate value property
Multivariable calculus
See Apostol (1985) ch12 & ch13, Rudin (1976) ch9
- LEC CAL1 15 ✅
- Partial and directional derivatives
- LEC CAL1 16 ✅
- Total derivatives, the Jacobian matrix
- LEC CAL1 17 ✅
- The chain rule
- LEC CAL1 18 ✅
- The mean value theorem for differentiable functions
- LEC CAL1 19 ✅
- Sufficient conditions for differentiability and equality of mixed partial derivatives
- LEC CAL1 20 ✅
- Taylor’s formula for functions from to
- LEC CAL1 21
- Inverse function theorem
- LEC CAL1 22
- Implicit function theorem
- LEC CAL1 23
- Extrema of real valued functions
- LEC CAL1 24
- Extremum problems with side conditions: Lagrange multipliers
References
Apostol, T. M. (1985). Mathematical Analysis (2d ed). Narosa.
Ghorpade, S., & Limaye, B. V. (2018). A Course in Calculus and Real Analysis (Second edition). Springer.
Royden, H. L., & Fitzpatrick, P. (2014). Real Analysis (4th edition, updated printing). Pearson.
Rudin, W. (1976). Principles of Mathematical Analysis (3d ed). McGraw-Hill.