302CRS Analysis 2

CMI, Aug-Nov 2025, R Srinivasan
Kumaresan (2005), Rudin (1976), Royden & Fitzpatrick (2014), Pugh (2015), Lecture recordings

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Lectures

Metric spaces

LEC ANA2 1 Sequence spaces: $l_p$ is a NLS
LEC ANA2 2 Examples of metric spaces

Completeness, Separability, Compactness, Equicontinuity

LEC ANA2 3 ✅ $B(S)$, $C_b(S)$ are Banach spaces, some comments about $C^1[0,1]$
LEC ANA2 4 ✅ Completeness of $l_1$, uniqueness of completion
LEC ANA2 5 ✅ Alternate construction of completion, separability
LEC ANA2 6 ✅ Second countability, compactness
LEC ANA2 7 ✅ Equicontinuity, Arzelà–Ascoli theorem

Baire’s Theorem

LEC ANA2 8 ✅ Banach’s Contraction principle, Baire Category theorem
LEC ANA2 9 ✅ Continuous nowhere differentiable functions are second category in $C[0,1]$.
LEC ANA2 10 ✅ More applications of Baire’s theorem: Discontinuities of pointwise limit of continuous functions, Uniform boundedness theorem

Stone-Weierstrass Theorem

LEC ANA2 11 ✅ Stone-Weierstrass for $\mathbb{R}$ and $\mathbb{C}$
LEC ANA2 12 ✅ Stone-Weierstrass for locally compact metric spaces

Connectedness

LEC ANA2 13 Connectedness
LEC ANA2 14
LEC ANA2 15

Definition 7.1(Definition).

A topological space is called

1. Hausdorff if distinct points have disjoint open neighborhoods.
2. regular if every closed set $C$ and point $p\notin C$ have have disjoint open neighborhoods.
3. normal if any two disjoint closed sets have disjoint open neighborhoods.

A normal space need not be Hausdorff in general.

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AS ANA2 1
AS ANA2 2

TST ANA2 Quiz 1
TST ANA2 Quiz 2

TUT ANA2 1

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References

Kumaresan, S. (2005). Topology of Metric Spaces. Alpha Science International Ltd.

Pugh, C. C. (2015). Real Mathematical Analysis. Springer International Publishing. https://doi.org/10.1007/978-3-319-17771-7

Royden, H. L., & Fitzpatrick, P. (2014). Real Analysis (4th edition, updated printing). Pearson.

Rudin, W. (1976). Principles of Mathematical Analysis (3d ed). McGraw-Hill.