ε Analysis & Topology

102 topics

• Axioms of the Real Numbers (Expert) – ℝ is a complete ordered field.
• Completeness and Supremum (Expert) – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property (Expert) – For any real x there is an integer n > x.
• Density of Rationals (Expert) – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) (Expert) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences (Expert) – Algebra of limits and squeeze theorem.
• Monotone Convergence Theorem (Expert) – Bounded monotone sequences converge.
• Subsequences (Expert) – Sequences extracted from sequences; every limit point is a subsequential limit.
• Bolzano–Weierstrass Theorem (Expert) – Every bounded sequence has a convergent subsequence.
• Cauchy Sequences (Expert) – Terms get arbitrarily close to each other.
• Limsup and Liminf (Expert) – Upper and lower limits of sequences.
• Series Convergence (Analysis) (Expert) – Partial sums converge to a limit.
• Rearrangement of Series (Expert) – Conditionally convergent series can be rearranged to any sum (Riemann).
• Limits of Functions (ε–δ) (Expert) – Formal definition of a limit.
• Continuity (ε–δ) (Expert) – Formal continuity at a point.
• Uniform Continuity (Expert) – δ depends only on ε, not the point.
• Extreme Value Theorem (Expert) – A continuous function on [a,b] attains its max and min.
• Differentiability and Continuity (Expert) – Differentiable implies continuous but not the converse.
• Weierstrass Function (Expert) – A continuous function nowhere differentiable.
• Darboux’s Theorem (Expert) – Derivatives have the intermediate value property.
• Riemann Integral (Expert) – Defined by upper and lower Darboux sums.
• Integrability Criteria (Expert) – Continuous or monotone functions are integrable.
• Pointwise vs Uniform Convergence (Expert) – Convergence of function sequences.
• Weierstrass M-Test (Expert) – Test for uniform convergence of series.
• Uniform Limits and Continuity (Expert) – Uniform limits of continuous functions are continuous.
• Power Series Radius of Convergence (Expert) – Cauchy–Hadamard formula.
• Taylor’s Theorem with Remainder (Expert) – Lagrange form of the error term.
• Analytic Functions (Expert) – Functions equal to their Taylor series locally.
• Stone–Weierstrass Theorem (Expert) – Polynomials are dense in continuous functions.
• Fourier Series Convergence (Expert) – Pointwise and L² convergence.
• Metric Spaces (Expert) – Sets with a distance function.
• Open and Closed Sets (Expert) – Basic building blocks of topology.
• Interior, Closure and Boundary (Expert) – Operations describing the position of points relative to a set.
• Compactness (Expert) – Every open cover has a finite subcover.
• Heine–Borel Theorem (Expert) – In ℝⁿ, compact means closed and bounded.
• Connectedness (Expert) – A space that cannot be split into two open sets.
• Completeness of Metric Spaces (Expert) – Every Cauchy sequence converges.
• Banach Fixed Point Theorem (Expert) – A contraction on a complete space has a unique fixed point.
• Baire Category Theorem (Expert) – Complete metric spaces are not countable unions of nowhere dense sets.
• Arzelà–Ascoli Theorem (Expert) – Compactness in function spaces.
• Topological Spaces (Expert) – Sets with a collection of open sets.
• Bases and Subbases (Expert) – Generating a topology from simple sets.
• Continuity in Topology (Expert) – Preimages of open sets are open.
• Homeomorphisms (Expert) – Continuous bijections with continuous inverses.
• Product and Quotient Topologies (Expert) – Combine or collapse spaces.
• Separation Axioms (Expert) – Hausdorff, regular and normal spaces.
• Urysohn’s Lemma (Expert) – Separate closed sets by continuous functions.
• Tychonoff’s Theorem (Expert) – Products of compact spaces are compact.
• Fundamental Group (Expert) – Loops up to deformation capture holes.
• Covering Spaces (Expert) – Spaces that locally look like copies of the base.
• Homology Introduction (Expert) – Algebraic counts of holes.
• Euler Characteristic (Expert) – A topological invariant of surfaces.
• Manifolds (Expert) – Spaces locally like ℝⁿ.
• Knot Theory Introduction (Expert) – Study of embeddings of circles in 3-space.
• Sigma-Algebras (Expert) – Collections of sets closed under complement and countable unions.
• Measures (Expert) – Functions assigning size to sets.
• Lebesgue Measure (Expert) – Generalises length and volume.
• Measurable Functions (Expert) – Functions compatible with measures.
• Lebesgue Integral (Expert) – Integrate by partitioning the range.
• Monotone Convergence (Lebesgue) (Expert) – Limits of increasing nonnegative functions.
• Dominated Convergence Theorem (Expert) – Swap limit and integral under a dominating function.
• Fatou’s Lemma (Expert) – Inequality for integrals of limits inferior.
• Fubini’s Theorem (Expert) – Iterated integrals equal double integrals.
• Lp Spaces (Expert) – Spaces of functions with integrable p-th power.
• Hölder and Minkowski Inequalities (Expert) – Key inequalities for Lp spaces.
• Hilbert Spaces (Expert) – Complete inner product spaces.
• Banach Spaces (Expert) – Complete normed vector spaces.
• Bounded Linear Operators (Expert) – Continuous linear maps between normed spaces.
• Hahn–Banach Theorem (Expert) – Extend linear functionals with norm control.
• Open Mapping and Closed Graph (Expert) – Pillars of functional analysis.
• Spectral Theory Introduction (Expert) – Generalising eigenvalues to operators.
• Complex Functions (Expert) – Functions of a complex variable f(z).
• Complex Differentiability (Expert) – Holomorphic functions.
• Cauchy–Riemann Equations (Expert) – Conditions for complex differentiability.
• Harmonic Functions (Expert) – Solutions of Laplace’s equation.
• Complex Exponential and Logarithm (Expert) – Multi-valued log in the complex plane.
• Contour Integrals (Expert) – Integrals of complex functions along paths.
• Cauchy’s Integral Theorem (Expert) – Integral around a closed loop of a holomorphic function is 0.
• Cauchy’s Integral Formula (Expert) – Values inside a loop are determined by boundary values.
• Liouville’s Theorem (Expert) – Bounded entire functions are constant.
• Fundamental Theorem of Algebra (Expert) – Proof via Liouville’s theorem.
• Laurent Series (Expert) – Series with negative powers around singularities.
• Singularities and Residues (Expert) – Poles, removable and essential singularities.
• Residue Theorem (Expert) – Integral equals 2πi times the sum of residues.
• Real Integrals by Residues (Expert) – Compute difficult real integrals using complex contours.
• Conformal Maps (Expert) – Angle-preserving transformations.
• Möbius Transformations (Expert) – Maps of the form (az+b)/(cz+d).
• Riemann Mapping Theorem (Expert) – Simply connected domains map conformally to the disk.
• Analytic Continuation (Expert) – Extend a function beyond its original domain.
• Riemann Zeta Function (Expert) – Links primes and complex analysis.
• Riemann Hypothesis (Expert) – Nontrivial zeros of ζ lie on Re(s) = ½.
• Mandelbrot and Julia Sets (Expert) – Fractals from iterating z² + c.
• Dynamical Systems and Iteration (Expert) – Fixed points, cycles and chaos.
• Lyapunov Exponents and Chaos (Expert) – Sensitivity to initial conditions.
• Cantor Set (Expert) – A perfect, uncountable set of measure zero.
• p-adic Numbers (Expert) – An alternative completion of the rationals.
• Hausdorff Dimension (Expert) – A fractional notion of dimension.
• Generalised Functions (Expert) – Delta function and weak derivatives.
• Fourier Transform (Expert) – Decompose functions into frequencies.
• Convolution (Expert) – Smooth or combine functions.
• Sobolev Spaces Intro (Expert) – Function spaces with weak derivatives.
• Real Analysis Problem-Solving (Expert) – Strategies for proofs about limits and continuity.