MTH301: (Real Analysis)

[Cr:4, Lc:3, Tt:1, Lb:0]

Review of basic concepts of real numbers: Archimedean property, completeness.
Metric spaces (definition and examples including spaces of sequences), compactness criteria in metric spaces and ${\mathbb{R}}^n$ in particular, connectedness in ${\mathbb{R}}^n$ .
Continuity, uniform continuity, monotonic functions, derivatives, Taylor’s theorem, Riemann integral and its properties.
Sequence and series of functions (with illustrative examples showing how the properties (like continuity, differentiability etc) of the limiting function need not be same as that of the properties of the sequence of functions), notion of interchange of limits in analysis, uniform convergence and its relation with continuity, differentiation and integration..
C([0,1]) as a metric space with the standard sup metric and characterisation of its compact sets/equicontinuous families of functions (Arzela Ascoli theorem), Weierstrass approximation theorem (at least the statement, with examples).
Functions in several variables, Inverse function theorem, Implicit function theorem, the rank theorem.

: Stone Weirestrass theorem, Fourier series