**Course Contents BS Mathematics 7th Semester**

Course Title:

**Functional Analysis I**

Credit
Hours: 3 + 0

**Metric Spaces**

- Review of metric spaces
- Convergence in metric spaces
- Complete metric spaces
- Completeness proofs
- Dense sets and separable spaces
- No-where dense sets

**Normed Spaces**

- Normed linear spaces
- Banach spaces
- Convex sets
- Quotient spaces
- Equivalent norms
- Linear operators
- Linear functional
- Finite dimensional normed space
- Continuous or bounded linear operators
- Dual spaces

**Inner Product Spaces**

- Definition and examples
- Orthonormal sets and bases
- Annihilators, projections
- Hilbert space
- Linear functional on Hilbert spaces
- Reflexivity of Hilbert spaces

*Recommended
Books***: **

^{}

- Curtain RF, Pritchard
AJ,
*Functional Analysis in Modern Applied Mathematics, Academic Press, New York*

- Friedman A,
*Foundations of Modern Analysis*, 1982, Dover Publications

- Kreyszig E,
*Introductory Functional Analysis with Applications*, John Wiley, New York - Rudin W,
*Functional Analysis*, 1973, McGraw Hill, New York

**Course Contents BS Mathematics 5th (E)**

Course Title:

**Algebra II**Course Code: MATH 301 Credit
Hours: 3 + 0

**Groups**

- Definition and examples of groups
- Abelian group
- Subgroups lattice, Lagrange’s theorem
- Relation between groups
- Cyclic groups
- Groups and symmetries, Cayley’s theorem

**Complexes in Groups**

- Complexes and coset decomposition of groups
- Centre of a group
- Normalizer in a group
- Centralizer in a group
- Conjugacy classes and congruence relation in a group

**Normal Subgroups**

- Normal subgroups
- Proper and improper normal subgroups
- Factor groups
- Fundamental theorem of homomorphism

**Sylow Theorems**

- Cauchy’s theorem for Abelian and non-Abelian group
- Sylow theorems

**Ring Theory**

- Definition and example of rings
- Special classes of rings
- Fields
- Ideals and quotient rings
- Ring homomorphisms
- Prime and maximal ideals
- Field of quotients

*Recommended Books***:**

- Allenby RBJT,
*Rings, Fields and Groups: An Introduction to Abstract Algebra*, 1983, Edward Arnold - J. B. Fraleigh,
*A First Course in Abstract Algebra*, 7^{th}edition, (Addison-Weseley Publishing Co., 2003) - Macdonald ID,
*The Theory of Groups,*1975, Oxford Clarendon Press, Ma., USA - P. B. Bhattacharya,
S. K. Jain and S. R. Nagpaul,
*Basic Abstract Algebra*, (Cambridge University Press, 1986) - Vijay K. Khanna, S K
Bhambri, A Course in Abstract Algebra(2
^{nd}Revised Edition) Vikas Publishing House PVT LTD. - H. Marshall,
*The Theory of Groups*, (Macmillan, 1967) - Humphreys, J.F.:
*A Course in Group Theory*(Oxford University Press, 2004) - Lederman, W.:
*Introduction to Group Theory*(Cambridge University Press, 1987). - Burton, D.E.:
*A First Course in Rings and Ideals*(Addision Wesley Pub. Co., 1968).