I. Connected Metric Spaces:
1. Definition, connex subspaces of R, theorems and properties
2. Continuity et connexity, Intermediate value theorem, Product space.
3. Path-Connected Spaces : definitions et properties
1. Hermitian Form, Cauchy-Schwarz’ Inequqlity, Inner Product, préhilbert Spaces.
2. Orthogonality- theoreme of Pythagore, orthogonal projection.
3. Topological Dual of Hilbert Spaces, Representation Theorem of Riez.
4. Orthogonal Systems – Bessel’s Inequality, Parseval’s Inequality
1. Definition: Stairs Function and its Integral.
2. Integrable function in the sense of Riemann- properties et theorems.
3. Piecewise Continuity.
4. Indefinite Integral – Antiderivative.
II. Complex Analysis:
Continuity, differentiability, elementary functions, complex integrals, Cauchy’s formula, Power series, isolated singular points, Residue and Poles, Integration.