Real analysis for the undergraduate: with an invitation to functional analysis / Matthew A. Pons

Includes bibliographical references and index.

The real numbers -- Preliminaries -- Complete ordered fields -- The real number system -- Set structures in R -- Normed linear spaces -- Sequences in R -- Sequences and convergence -- Properties of convergent sequences -- Completeness in R revisited -- Set structures in R via sequences -- Complete spaces -- Numerical series -- Series of real numbers -- Basic convergence tests -- Absolute and conditional convergence -- Sequence spaces -- Continuity -- Sequences and the limit of a function -- Continuity -- The intermediate value theorem -- Continuity on a set and uniform continuity -- Bounded linear operators -- The derivative -- The definition of the derivative -- Properties of the derivative -- Value theorems for the derivative -- Consequences of the value theorems -- Taylor polynomials -- Eigenvalues and the invariant subspace problem -- Sequences and series of functions -- Sequences of functions -- Series of functions -- Power series -- A continuous nowhere differentiable function -- Spaces of continuous functions -- The Riemann integral -- The Riemann integral -- Properties of the Riemann integral -- The Fundamental Theorem of Calculus -- The exponential function -- Spaces of continuous functions revisited -- Lebesgue measure on R -- Length and measure -- Outer measure on R -- Lebesgue measure on R -- A nonmeasurable set -- General measure theory -- Lebesgue integration -- Measurable functions -- The Lebesgue integral -- Limits and the Lebesgue integral -- The Lp spaces.