MA270 Analysis 3 - O-Umlaut

**Summary**: ... # 1. Introduction (Week 1) | Definitions | Theorems | Examples | | ------------------------------------------------------------ | ---------------------------------------------------------------------------------- | -------- | | 1.1 [[Convergence]] | [[Uniqueness of Limit of Convergent Sequence]]. | | | | [[Shift Rule for Limits]]. | | | | [[Convergent Real Sequence is Bounded]]. | | | | [[Convergence of Sequence implies Absolute Convergence of Sequence]]. | | | | [[Algebra of Limits of Convergent Sequences]]. | | | | [[Limits of Real Sequence Preserve Weak Inequalities]]. | | | | [[Sandwich Rule]]. | | | 1.2 [[Continuous Real Function]]. | | | | 1.3 [[Uniformly Continuous Real Function]]. | 1.4 [[Continuous Real Function on Closed Real Interval is Uniformly Continuous]]. | | | 1.5 [[Fréchet Differentiation]]. | 1.6 [[Weirestrass-Caratheodory Criterion for Differentiability of Real Function]]. | | | | | | | 1.7 [[Riemann integration]] & [[Riemann integration]]. | | | | 1.8 [[Riemann integration]] & [[Riemann integration]]. | | | | 1.9 [[Riemann integration]]. | 1.10 [[Riemann's criterion for integrability]]. | | | | 1.11 [[Continuous real functions are Riemann Integrable]]. | | | | 1.12 [[Linearity of Riemann integration]] | | | | 1.13 [[Monotonicity of Riemann Integral]]. | | | | 1.14 [[Triangle Inequality for Riemann Integral]]. | | | | 1.15 [[Fundamental theorem of calculus]]. | | | | 1.16 [[Fundamental theorem of calculus]]. | | # 2. Sequences and Series of Functions (Week 2 - Week 3) | Examples | Theorems | Definitions | | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------- | | [[Limit of Sequence of Real Functions is Non Commutative]]. [[Pointwise Convergence of Sequence of Real Functions doesn't Imply Uniform Convergence]]. [[Integral of Integrable Pointwise Limit of Sequence of Real Functions May not equal Pointwise Limit of Sequence of Integrals]]. [[Riemann-Lebesgue Lemma]]. [[Integral Success but Pointwise Failure]]. [[Pointwise Limit of Sequence of Differentiable Real Functions May be Continuous but Not Differentiable]]. | | 2.1. [[Pointwise Convergence of Sequence of Real Functions]]. | ### 2.2 Uniform Convergence | | | | | ------------------------------------------------------------------------------------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------- | | **Uniform convergence** of differentiable functions $f_n\to f$ does **not** imply $f$ is differentiable. [[Supremum Norm is a Norm]]. | 2.11. [[Sequence of Real Functions is Uniformly Convergent iff Uniformly Cauchy]]. 2.13 [[Uniform Limit of Sequence of Continuous Real Function is Continuous]]. 2.14 [[The Space of Continuous & Bounded Real Functions is Complete wrt Supremum Norm]]. 2.16 [[Uniform Convergence preserves Riemann Integrability]]. 2.17 [[Uniform Convergence and Differentiability]]. | 2.8. [[Uniform Convergence of Sequence of Real Functions]]. 2.10. [[Uniformly Cauchy Sequence of Real Functions]]. | ### 2.3 Series of Functions | | | | | ---------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------- | | | 2.19 [[Integral of Uniformly Convergent Series of Real Functions equals Series of Their Integrals]]. 2.20 [[Uniform Convergence and Differentiabilty for Series of Real Functions]]. 2.21 [[Comparison Test for Uniform Convergence of Series of Real Functions (Weierstrass M-test)]]. | 2.18 [[Pointwise Convergence of Series of Real Functions]]. 2.18 [[Uniform Convergence of Series of Real Functions]]. | # 3. $\varepsilon$ Convergence & Continuity in $\mathbb{R}^n$ *(Week 3 - Tuesday, Week 4)* | Examples | Theorems | Definitions/ Lemmas | | ------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------- | | 3.2 [[Comparison of Euclidean Norm with Max-Norm & Tax Cab Norm]]. | [[Euclidean spaces are normed spaces]]. [[Reverse triangle inequality]]. | [[Euclidean Norm]]. [[Angle Between Nonzero Real Vectors]]. | | | 3.7. [[Uniqueness of Limit of Convergent Sequence]]. 3.12. Algebra of Limits of Sequences follows from 3.8. 3.14 Boundedness of Convergent Sequences in $\mathbb{R}^n$ follows from 3.8 or 3.15. 3.15. Norm of Convergent Sequence converges to Norm of Limit which follows from Reverse triangle inequality. 3.17 [[Euclidean Space is a Complete Metric Space]]. 3.18 [[Bolzano-Weierstrass theorem]]. | 3.6. [[Convergence]]. 3.8. [[Equivalence of Component-wise Convergence and Convergence for Sequences in Euclidean Space]]. | ### 3.5 Continuity | Examples | Theorems | Definitions | | ------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------- | | | 3.4 [[Equivalence of Continuous and Sequentially Continuous Functions Over Euclidean Spaces]]. | 3.19 [[Continuous maps]]. 3.20 [[Sequentially Continuous Functions]]. 3.22 Limit | | 3.24 [[Separately Continuous Function may be Discontinuous]]. | 3.5 [[Continuous Functions are Separately Continuous]]. | 3.23 [[Separately Continuous Functions]]. | | | 3.25, 3.26, 3.7 [[Algebra of Continuous Functions Over Euclidean Spaces]]. 3.28 [[Composition of Continuous Functions in Continuous]]. 3.29 [[Continuous Function has Component-wise Continuity]]. | | | | 3.32. [[Continuity of Real-Valued Function of Several Real Variables]]. | 3.31 [[Continuity of Projections of Euclidean Space]]. | | | | 3.34 [[Linearly Continuous Functions]]. | # 4. Continuity via Open/ Closed Sets & Sequential Compactness (Tuesday, Week 4 - Tuesday, Week 5) ### 4.1. Closed & Open Subsets of $\mathbb{R}^n$ | Examples | Theorems | Definitions/ Lemmas | | ------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------- | | 4.6 [[Open Euclidean Ball is Open]] 4.8 [[Closed Euclidean Ball is Closed]]. 4.11 Enumeration of rationals ... | 4.3 [[Characterization of Open and Closed Subsets of Euclidean Space by Set Complement]]. 4.9 [[Union of Open Sets is Open]]. 4.12 [[Finite Intersection of open subsets of metric spaces are open]]. 4.13 [[Intersection of Closed Sets is Closed]] & [[Finite Union of Closed Sets is Closed]] | 4.1 [[Closed subsets of metric spaces]] 4.2 [[Open subsets of metric spaces]]. 4.5 [[Open Euclidean Ball]]. 4.7 [[Closed Euclidean Ball]]. | ### 4.2. Continuity & Topology | Examples | Theorems | Definitions | | ----------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------ | | 4.16 [[Image of open subset of Euclidean space under continuous function need not be open]]. 4.17 Unit sphere is closed. | 4.15 [[Characterization of Continuity Via Open Sets]]. | | | [[Unit n-Sphere is Sequentially Compact]]. Closed Euclidean Ball is sequentially compact. 4.25 Apply EVT to Euclidean Norm. | 4.20 [[Subset of Euclidean space is sequentially compact iff closed and bounded]]. 4.22 [[Continuous Functions Preserve Sequentially Compact Subsets of Euclidean Space]]. 4.23 [[Extreme Value Theorem]]. | [[Sequentially compactness]]. | # 5. The Space of Linear Maps (Tuesday, Week 5 - Monday, Week 6) ### 5.1. Two norms on the space of linear maps and matrices | Examples | Theorems | Definitions/ Lemmas | | -------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------- | | | 5.1.1 [[Frobenius & Operator Norms are Comparable]]. 5.1.2 [[Operator norm]]. 5.4. [[Operator Norm of Composition is Bounded by Product of Operator Norms]]. 5.5. [[Characterization of Injective Linear Maps via Euclidean Norm Inequality]]. 5.7 [[Perturbation of Injective Linear Map is Injective]] | 5.2 [[Operator norm]] | ### 5.2. Convergence and Continuity in $L(\mathbb{R}^n, \mathbb{R}^k)$ | Examples | Theorems | Definitions/ Lemmas | | -------- | -------- | ------------------- | | | | | | | | | ### 5.3 The Space $\text{GL}(n,\mathbb{R})$ of Invertible Linear Operators | Examples | Theorems | Definitions/ Lemmas | | -------- | -------- | ------------------- | | | | | | | | | # 6. The Derivative (Tuesday, Week 6 - Monday, Week 7) | Examples | Theorems | Definitions/ Lemmas | | -------- | -------- | ------------------- | | | | | # 7. Complex Analysis (Tuesday, Week 7 - Week 8) ### 7.1 Basic Facts About $\mathbb{C}$ | Examples | Theorems | Definitions | | -------- | -------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | | 7.7 [[Cauchy-Riemann Equations]]. 7.8 | 7.1 Converegent sequence in $\mathbb{C}$. 7.2 Open & closed sets in $\mathbb{C}$. 7.3 Sequential compactness of subset of $\mathbb{C}$. 7.4 Continuity of $f:\Omega \subset \mathbb{C} \to \mathbb{C}$. 7.5 [[Complex Differentiability]] 7.6 Analytic (holomorphic) functions | ### 7.2 Power Series | Examples | Theorems/ Lemmas | Definitions | | ----------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------- | | 7.16 | 7.11 [[Ratio Test for Series]]. 7.12 [[Cauchy's root test for absolute convergence]]. 7.13 [[Radius of Convergence of Complex Power Series]]. 7.14 Application of ratio test to radius of convergence. 7.15 [[Power Series is Termwise Differentiable within Radius of Convergence]] (proof is non-examinable). 7.17 [[Power Series Converges Uniformly within Radius of Convergence]] (similar to Weiestrass-M test) | 7.9 Convergence of series in $\mathbb{C}lt;br> 7.10 Absolute convergence of series in $\mathbb{C}$ | | 7.19 Trig-exp-hyperbolic relations. | 7.20 | 7.18 Power series expansion of exponential, trigonometric & hyperbolic functions. | | [[Complex Logarithm]]. | 7.21 [[Argument of Complex Number]]. | | ### 7.3 Complex Integration, Contour Integrals | Examples | Theorems | Definitions | | -------- | ----------------------------------------------------------------------------------------------------------------------- | -------------------------- | | | 7.24 [[Contour Integral along Reversed Curve]] & [[Contour Integral of Directed Curve is Reparametrisation Invariant]]. | 7.22 [[Contour Integral]]. | | | 7.28 [[Fundamental Theorem of Calculus for Contour Integrals]]. | | **7.31 Links with Green's and Divergence Theorems** | Examples | Theorems | Definitions | | --------------------------------- | ------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Fundamental Contour Integral]]. | 7.29 [[Cauchy-Goursat Theorem]]. 7.31 [[Principle of Deformation of Contours]]. | 7.30 [[Simple connectedness]]. | | | 7.33 [[Cauchy's Integral Formula]]. 7.35 [[Extension of Cauchy's Integral Formula]]. | 7.32 notation for interior and exterior of cure $\gamma$ are $I(\gamma)$ and $O(\gamma)$ so that $O(\gamma)$ is on your right ($\gamma$ is positively oriented). | **7.32 Consequences of Cauchy's Theorem** | Examples | Theorems | Definitions | | -------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------- | | | 7.36 [[Taylor's theorem]]. 7.38 [[Louiville's Theorem (Every Bounded Entire Function is Constant)]]. 7.39 [[Fundamental Theorem of Algebra]]. 7.40 [[Uniform Limit of Analytic Functions is Analytic]]. | | **7.33 Computing Integrals in $\mathbb{R}$ with Cauchy’s Integral Formula** See [[Evaluating Improper Integrals Using Contour Integration]].