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Functional Analysis I

Course Contents BS Mathematics 7th Semester

Course Code:   MATH 401
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:

  1. Curtain RF, Pritchard AJ, Functional Analysis in Modern Applied Mathematics, Academic Press, New York
  1. Friedman A, Foundations of Modern Analysis, 1982, Dover Publications
  1. Kreyszig E, Introductory Functional Analysis with Applications, John Wiley, New York
  2. Rudin W, Functional Analysis, 1973, McGraw Hill, New York

Algebra II

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