Real Analysis & Differentiable Manifolds

The book entitled Real Analysis & contains eight chapters. This book is written for UG and PG students. It includes Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and ifferentiation. Equicontinuous families of functions. The Stone- Weierstrass theorem. Differentiation. The Contraction Principle. The Inverse Function Theorem. The Implicit Function Theorem. Partitions of unity. The space of tangent vectors at a point of Rn. Vector fields on open subsets of Rn. Topological manifolds. Differentiable manifolds. Real Projective space. Differentiable functions and mappings. Rank of a mapping. Immersion. Sub manifolds. Outer measure. Measurable sets and Lebesgue measure. A non-measurable set, Measurable functions, Littlewood's three principles. The Riemann integral. Lebesgue integral of a bounded function over a set of finite measure. Integral of a non-negative function. General Lebesgue integral. Convergence in measure. Differentiation of monotone functions. Functions of bounded variation. Differentiation of an integral. Absolute continuity. Convex functions. Lp-spaces. Holder and Minkowski inequality.