Courses / Mathematics / Real Analysis II: uniform limits, metric spaces and local calculus Mathematics · unit version 1 Real Analysis II: uniform limits, metric spaces and local calculus Twenty proof-centered lessons on function convergence, metric structure and rigorous finite-dimensional differentiation. Written arguments and counterexamples require mathematical review. Measure theory, Lebesgue integration, infinite-dimensional calculus and full functional analysis are later subjects; this sequence does not claim independently assessed course equivalence.
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What changes when the cutoff N in a sequence of functions may depend on x? Uniform convergence asks for a common accuracy guarantee across the domain.
Compare reasoning and find a refresher Dependence on x gives a pointwise guarantee. Uniform convergence requires a single N for each positive tolerance that works for every x in the domain.
Review quantified sequence limits
Why is being bounded insufficient for a real sequence to converge? Metric-space compactness and completeness have different hypotheses and conclusions.
Compare reasoning and find a refresher The sequence (−1)ⁿ is bounded but does not converge. It has convergent subsequences; a bounded real sequence always has one, whereas convergence of the whole sequence needs more.
Review subsequences and compactness
What should a derivative do to a small input displacement in several dimensions? The advanced differentiation lessons require a linear approximation with a quantified remainder.
Compare reasoning and find a refresher A derivative is a linear map A for which f(a+h)−f(a)−Ah is small compared with ||h|| as h tends to zero. Coordinate partial derivatives alone do not establish this approximation.
Review multivariable differentiability
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Analysis II: uniform control, spaces and local theorems
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Lesson details and objectives Distinguish a cutoff chosen after a point from one controlling an entire domain.
Distinguish a cutoff chosen after a point from one controlling an entire domain. Compute uniform errors on a stated domain. Construct moving points that disprove a shared cutoff. 6 practice tasks.
Lesson details and objectives Prove completeness of bounded-function spaces from completeness of the scalar field.
Prove completeness of bounded-function spaces from completeness of the scalar field. Use the supremum norm only where it is finite. Prove both directions of the uniform Cauchy criterion. 6 practice tasks.
Lesson details and objectives Use a three-term estimate and distinguish limit regularity from convergence mode.
Use a three-term estimate and distinguish limit regularity from convergence mode. Prove continuity of a uniform limit at an arbitrary point. Separate sufficient uniformity from necessary continuity of a limit. 6 practice tasks.
Lesson details and objectives Control integral errors by uniform error and exhibit concentration that defeats pointwise passage.
Control integral errors by uniform error and exhibit concentration that defeats pointwise passage. Prove integrability of a uniform limit on a compact interval. Explain why a finite-domain uniform estimate cannot justify an infinite-tail interchange. 6 practice tasks.
Lesson details and objectives Use derivative convergence plus one anchor value to recover a differentiable limit.
Use derivative convergence plus one anchor value to recover a differentiable limit. Prove a derivative-interchange theorem under explicit C¹ hypotheses. Construct smooth approximations whose limit loses differentiability. 8 practice tasks.
Lesson details and objectives Dominate every pointwise tail by one convergent numerical series.
Dominate every pointwise tail by one convergent numerical series. Prove the M-test using the uniform Cauchy criterion. Distinguish uniform convergence of a series from uniform convergence of its derivative series. 6 practice tasks.
Lesson details and objectives Prove interior uniform control and justify termwise calculus inside the convergence radius.
Prove interior uniform control and justify termwise calculus inside the convergence radius. Derive a geometric majorant on a smaller closed interval. Explain why endpoint convergence needs a separate argument. 8 practice tasks.
Lesson details and objectives Test distance axioms and build open balls without assuming Euclidean coordinates.
Test distance axioms and build open balls without assuming Euclidean coordinates. Verify standard and discrete metrics. Prove balls are open and distinguish relative topology from ambient topology. 6 practice tasks.
Lesson details and objectives Transport epsilon arguments to general metric spaces and characterize continuity by inverse images.
Transport epsilon arguments to general metric spaces and characterize continuity by inverse images. Prove sequential continuity and its converse in metric spaces. Prove the inverse-image characterization without assuming coordinates. 6 practice tasks.
Lesson details and objectives Separate pairwise tail control from existence of a limit in the chosen space.
Separate pairwise tail control from existence of a limit in the chosen space. Prove closed subspaces of complete spaces are complete. Construct Cauchy sequences that expose a missing point. 6 practice tasks.
Lesson details and objectives Distinguish finite approximation at every scale from ordinary boundedness.
Distinguish finite approximation at every scale from ordinary boundedness. Construct finite epsilon nets on intervals. Extract Cauchy subsequences by nested infinite selections. 6 practice tasks.
Lesson details and objectives Prove the metric compactness equivalences rather than importing Heine–Borel blindly.
Prove the metric compactness equivalences rather than importing Heine–Borel blindly. Show compact metric spaces are complete and totally bounded. Prove a complete totally bounded metric space is compact. 6 practice tasks.
Lesson details and objectives Use separations and inverse images to prove global intermediate-value behavior.
Use separations and inverse images to prove global intermediate-value behavior. Prove continuous images of connected spaces are connected. Distinguish connectedness from path connectedness without claiming a converse. 6 practice tasks.
Lesson details and objectives Prove convergence, existence, uniqueness and a computable tail bound under contraction hypotheses.
Prove convergence, existence, uniqueness and a computable tail bound under contraction hypotheses. Derive geometric iterate error bounds. Identify failures caused by missing completeness or strict contraction. 6 practice tasks.
Lesson details and objectives Measure linear maps uniformly over directions and justify finite-dimensional norm equivalence.
Measure linear maps uniformly over directions and justify finite-dimensional norm equivalence. Compute and bound induced operator norms. Use compactness of a Euclidean sphere to compare arbitrary norms. 6 practice tasks.
Lesson details and objectives Require a remainder small relative to displacement in every direction.
Require a remainder small relative to displacement in every direction. Prove uniqueness of the derivative map. Distinguish directional derivatives from total differentiability. 6 practice tasks.
Lesson details and objectives Compose derivative maps while controlling how an inner remainder changes scale.
Compose derivative maps while controlling how an inner remainder changes scale. Prove the chain rule using small-o remainders. Use derivative bounds on convex regions to obtain Lipschitz estimates. 8 practice tasks.
Lesson details and objectives Reduce finite-dimensional Taylor estimates to one variable along a segment.
Reduce finite-dimensional Taylor estimates to one variable along a segment. Derive the integral second-order remainder. Use Hessian definiteness to classify nondegenerate critical points. 8 practice tasks.
Lesson details and objectives Use a local contraction to explain why an invertible derivative gives a local inverse.
Use a local contraction to explain why an invertible derivative gives a local inverse. State the C¹ local inverse theorem with open neighborhoods. Separate local invertibility from global injectivity. 6 practice tasks.
Lesson details and objectives Solve constraints locally using an invertible derivative block and derive sensitivities.
Solve constraints locally using an invertible derivative block and derive sensitivities. Reduce the implicit theorem to a local inverse map. Compute the derivative of a locally determined variable and diagnose singular points. 6 practice tasks.
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