Real Analysis II
A rigorous second course in analysis that moves from numbers to functions and from one variable to several: why limits of functions matter and how pointwise limits lose continuity, integrals and derivatives, from Cauchy's 1821 error and Abel's 1826 counterexample to Weierstrass's uniform convergence, the sup norm and the uniform Cauchy criterion, continuity, integration and differentiation of uniform limits and Dini's theorem, series of functions and the Weierstrass M-test, power series, radius of convergence, Abel's theorem and analytic functions, the Weierstrass approximation theorem proved with Bernstein polynomials, metric spaces with their open and closed sets, completeness of function spaces, compactness and the Heine–Borel theorem, Banach's contraction mapping theorem, continuity on compact spaces and an overview of the Arzelà–Ascoli theorem, the derivative in Rⁿ as a linear map with the chain rule and the mean value inequality, the inverse function theorem, and a capstone on the implicit function theorem, curves and surfaces defined by equations and Lagrange multipliers, with complete proofs and runnable Python examples throughout.
MATH-341 Year 3 3 credits 12 lessons · 4 h 59 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Explain why limits of functions matter, give examples in which pointwise limits lose continuity, integrals or derivatives, and describe the history from Cauchy’s 1821 error and Abel’s 1826 counterexample to Weierstrass’s uniform convergence.
- Define pointwise and uniform convergence, test for uniform convergence with the sup norm, disprove it with well-chosen points, and prove and apply the uniform Cauchy criterion.
- Prove that uniform limits of continuous functions are continuous, that uniform convergence on a closed bounded interval allows limits and integrals to be interchanged, and that term-by-term differentiation is valid when the derivatives converge uniformly; prove and apply Dini’s theorem.
- Analyze series of functions with the Cauchy criterion and the Weierstrass M-test, use its tail bound, and explain the continuous nowhere differentiable function of Weierstrass.
- Find radii of convergence by the Cauchy–Hadamard and ratio formulas, prove that power series converge uniformly on compact subintervals and may be differentiated and integrated term by term, prove Abel’s theorem, and distinguish smooth from analytic functions.
- State and prove the Weierstrass approximation theorem by Bernstein polynomials, estimate the error, and apply the theorem to moments and density arguments.
- Define metric spaces, verify the axioms for standard examples including the Euclidean, taxicab and sup metrics and the sup and integral metrics on spaces of functions, and work with open balls, open and closed sets, closures and equivalent metrics.
- Prove completeness of the real spaces of several variables and of C[a, b] with the sup metric, show that C[0, 1] with the integral metric is incomplete, prove that compactness and sequential compactness agree in metric spaces, and prove the Heine–Borel theorem.
- State and prove the Banach fixed point theorem with its error bounds, recognize contractions by the mean value theorem, and use iteration to solve equations and integral equations.
- Characterize continuity by sequences and by open sets, prove that continuous maps on compact spaces are bounded, attain extreme values and are uniformly continuous, and state, prove and apply the Arzelà–Ascoli theorem.
- Define the derivative of a map between spaces of several variables as a linear map, compute Jacobian matrices, prove that continuous partial derivatives imply differentiability, and prove the chain rule and the mean value inequality.
- State and prove the inverse and implicit function theorems, compute derivatives of inverse and implicit functions, and apply them to local invertibility, curves and surfaces defined by equations, and Lagrange multipliers.
Course content
12 lessons · 4 h 59 min total · final exam
- 1 From numbers to functions: why convergence of functions matters 27 min · quiz
- 2 Pointwise and uniform convergence 26 min · quiz
- 3 Uniform convergence, continuity, integration and differentiation 24 min · quiz
- 4 Series of functions and the Weierstrass M-test 26 min · quiz
- 5 Power series and analytic functions on the real line 21 min · quiz
- 6 The Weierstrass approximation theorem 24 min · quiz
- 7 Metric spaces: examples and open and closed sets 23 min · quiz
- 8 Completeness, compactness and the contraction mapping theorem 24 min · quiz
- 9 Continuity in metric spaces and Arzelà–Ascoli (overview) 23 min · quiz
- 10 Differentiation in Rⁿ: the derivative as a linear map 27 min · quiz
- 11 The inverse function theorem 23 min · quiz
- 12 Capstone: the implicit function theorem and applications 31 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A rigorous second course in analysis that moves from numbers to functions and from one variable to several: why limits of functions matter and how pointwise limits lose continuity, integrals and derivatives, from Cauchy's 1821 error and Abel's 1826 counterexample to Weierstrass's uniform convergence, the sup norm and the uniform Cauchy criterion, continuity, integration and differentiation of uniform limits and Dini's theorem, series of functions and the Weierstrass M-test, power series, radius of convergence, Abel's theorem and analytic functions, the Weierstrass approximation theorem proved with Bernstein polynomials, metric spaces with their open and closed sets, completeness of function spaces, compactness and the Heine–Borel theorem, Banach's contraction mapping theorem, continuity on compact spaces and an overview of the Arzelà–Ascoli theorem, the derivative in Rⁿ as a linear map with the chain rule and the mean value inequality, the inverse function theorem, and a capstone on the implicit function theorem, curves and surfaces defined by equations and Lagrange multipliers, with complete proofs and runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.