تحليل حقيقي - REAL ANALYSIS

The Riemann-Stieltjes Integral: Definition and existence of the integral, properties of the integral, integration and differentiation, integration of vector-valued function, rectifiable curves. Normed spaces, Banach spaces and their examples, Examples of incomplete normed spaces, Subspace of normed spaces, Isometry on normed spaces, Completion of normed linear spaces, Quotient spaces, Product spaces, Schauder basis, Infinite series in normed space: convergence and absolute convergence, Finite dimensional normed spaces, Equivalent norms, Compactness, Riesz Lemma, Denseness and separability properties. Bounded linear operators and bounded linear functionals with their norms and properties, Unbounded linear operators, Space of bounded linear operators, Dual basis, Algebraic and topological duals and relevant results, Duals of some standard normed spaces, Inner Product Space, Hilbert Space, Further Properties of Inner Product Spaces.