Real Analysis

 This course offers basic concepts of Real Analysis

Basic Topology 2025-26

This course introduces the foundational concepts of general topology, a central area of modern mathematics with applications across analysis and geometry. Students will study topological spaces, continuity, open and closed sets, convergence, compactness, and connectedness. Emphasis is placed on understanding various classes of spaces, separation axioms, and function properties in topological contexts. The course fosters abstract thinking and lays the groundwork for advanced studies in mathematics and theoretical computer science.

Course Outcomes (COs):

  • CO1: Understand the structure and examples of topological spaces, including bases and subbases.

  • CO2: Analyze closure, interior, neighbourhood systems, and function continuity in topological spaces.

  • CO3: Evaluate topological properties such as compactness and connectedness, and explore smallness conditions.

  • CO4: Interpret and apply advanced separation axioms and properties of locally connected and path-connected spaces.


Functional Analysis 2025-26

This course offers a structured introduction to the core concepts of Functional Analysis, focusing on the study of metric, normed, and inner product spaces, along with linear operators and functionals. It explores foundational results such as the Hahn–Banach theorem, properties of Hilbert spaces, and the theory of bounded operators. Emphasis is placed on abstract reasoning and rigorous mathematical proofs, which are essential in pure mathematics and applications like quantum mechanics, differential equations, and optimization.

Course Outcomes (COs):

  • CO1: Understand the structure and properties of metric and normed linear spaces and recognize complete (Banach) spaces.

  • CO2: Analyze linear operators and functionals, and interpret their roles in normed spaces and dual spaces.

  • CO3: Apply the properties of inner product spaces and Hilbert spaces in mathematical and applied contexts.

  • CO4: Interpret and utilize key functional analytic theorems (e.g., Hahn-Banach, Zorn’s Lemma) and understand the behavior of adjoint and normal operators in Hilbert spaces.