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Course

Code:	Math 4
Subject:	Mathematics
Topic:	Real Analysis
Views:	56,433

Educator

Name:	Technical University of Denmark (DTU)
Type:	University

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Advanced Real Analysis with Ole Christensen

Video Lectures

Displaying all 25 video lectures.

Lecture 1 Play Video	Normed Vector Spaces I Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 06:45 - Vector Spaces; 07:15 - Example 1; 12:00 - Mathematical Tool - Fourier Transform; 17:00 - Example 2; 20:00 - Example 3; 23:00 - New Concept - Norm; 27:45 - Lemma 2.1.2 - The Opposite Triangle Inequality; 35:15 - Convergence; 39:30 - Exact Definition; 41:45 - Goal; 43:15 - Subspaces; 44:15 - Characterisation Of A Subspace; 46:15 - Example: A Trigonometric Polynomial;
Lecture 2 Play Video	Normed Vector Spaces II Lecture with Ole Christensen. Kapitler: 00:00 - Boundedness/Supremum; 05:00 - Example; 08:00 - Maximum Value; 09:00 - Example: Sup Vs. Max; 12:45 - Theorem: Maximum Is Attained On Closed And Bounded Intervals; 15:30 - Vectorspace Of Continuous Functions; 22:00 - Norm On C[A,B]; 36:45 - Example Of Convergence In C[A,B]; 49:45 - Closing Remarks;
Lecture 3 Play Video	Banach Spaces I Lecture with Ole Christensen. Kapitler: 00:00 - Banach Spaces; 06:30 - Cauchy Sequences; 12:00 - Def: Banach Space; 15:45 - Examples; 17:15 - C[A,B] Is Banach With Proof; 36:30 - Ex: Sequence Space L^1(N); 46:45 - Sequence Space L^p(N);
Lecture 4 Play Video	Banach Spaces II Lecture with Ole Christensen. Kapitler: 00:00 - More On L^p(N); 06:30 - Minkowski Inequality; 08:30 - Linear Operators; 13:00 - Bounded Linear Operators; 15:00 - Operator Norm; 19:45 - T:C[0,2]-→C[0,2],Tf(X)=X^2*F(X); 35:00 - Operator On L^p;
Lecture 5 Play Video	Hilbert Spaces I Lecture with Ole Christensen. Kapitler: 00:00 - Repetition; 03:45 - R^n Is Banach; 07:00 - Inner Product; 14:00 - Example: C^n; 22:45 - What About ←V,Aw+Bu→; 25:30 - R^2; 28:15 - Cauchy Schwarz Inequality; 30:15 - Inner Product Induces A Norm; 41:30 - Inner Product On Real Spaces; 43:45 - Important Properties Of An Inner Product;
Lecture 6 Play Video	Hilbert Spaces II Lecture with Ole Christensen. Kapitler: 00:00 - Def: Hilbert Space; 05:00 - New Example Of A Hilbert Space; 15:15 - Operators On Hilbert Spaces; 20:00 - Example 1; 24:00 - Example 2; 38:30 - Riesz Representation Theorem; 43:00 - Concerning Physics;
Lecture 7 Play Video	Adjoint Operator I Lecture with Mads Jakobsen. Kapitler: 00:00 - Introduction; 00:30 - Homework; 04:30 - Normed Vector Spaces; 08:30 - The Adjoint Operator; 18:30 - Theorem 4.5.1; 19:30 - Proof; 24:00 - Lema 4.4.2; 32:30 - Example Week 2;
Lecture 8 Play Video	Adjoint Operator II Lecture with Mads Jakobsen. Kapitler: 00:00 - Explanation; 01:45 - Definition 4.5.4; 12:45 - Definition Inverse Of T; 13:45 - Exercise 4.19; 20:15 - Basis; 20:45 - Recall; 24:15 - Example; 37:00 - Example;
Lecture 9 Play Video	Lp Spaces on the real line Lecture with Ole Christensen. Kapitler: 00:00 - Repetition; 05:00 - Introduction; 15:00 - Inequalities For Integrals; 26:00 - Support Of A Function; 31:30 - Special Continuous Functions; 34:15 - Examples;
Lecture 10 Play Video	Lp Spaces On The Real Line II Lecture with Ole Christensen. Kapitler: 00:00 - Remarks On Banach Spaces; 08:00 - Proof That Cc Is Not A Banach Space; 31:00 - Applications; 38:30 - Integral Operators;
Lecture 11 Play Video	More On Lp And L2 Spaces I Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 05:00 - Complication With Norm On Lp Spaces; 11:30 - Dis/Advantages With Cc,Co,Lp Spaces; 20:00 - Why Not Only Use L1?; 22:00 - Cc Is Dense In L1; 26:00 - Recall Hilbert Spaces; 28:00 - L2 Is A Hilbert Space ; 39:00 - Operators On L2;
Lecture 12 Play Video	More On Lp And L2 Spaces II Lecture with Ole Christensen. Kapitler: 00:00 - The Translation Operator; 03:00 - The Modulation Operator; 05:45 - The Dilation Operator; 10:45 - Properties Of The Three Operators; 12:45 - Proof Of Properties For Translation Operator; 13:00 - Well Defined; 16:45 - Linear; 19:30 - Bounded; 23:00 - Unitary; 35:00 - The Momentum Operator;
Lecture 13 Play Video	More On Operators On L2 I Lecture with Ole Christensen. Kapitler: 00:00 - Repetition - L^2(R); 06:30 - Composition Of Opertors; 21:00 - Basis In Hilbert Spaces; 26:30 - Introduction To Orthonormal Bases; 29:30 - Def: Orthonormal System; 31:00 - Def: Orthonormal Basis; 33:15 - The Theorem 4.7.2 ; 43:00 - Proof Of Thrm 4.7.2;
Lecture 14 Play Video	Orthonormal Bases Vs Fourier Series II Lecture with Ole Christensen. Kapitler: 00:00 - Proof Of Thrm 4.7.2 Continued; 11:00 - Connection To Fourier Series; 11:15 - L2(-Pi,Pi); 16:00 - Complex Fourier Series; 17:45 - Convergence?; 35:45 - Parseval Identity;
Lecture 15 Play Video	Approximation Theory I Lecture with Ole Christensen. Kapitler: 00:00 - Intro To Approximation Theory; 10:00 - Remarks On Vectorspaces In Mat4; 13:30 - Def.: Dense Subset; 19:15 - Dense Subspace Of The Sequence Spaces L^p; 24:45 - Dense Subspace Of The Function Spaces L^p; 35:15 - Weierstrass Approximation Theorem;
Lecture 16 Play Video	Approximation Theory II Lecture with Ole Christensen. Kapitler: 00:00 - Def.: Closure Of A Subset; 06:45 - Dense Vs. Closure; 19:00 - Extension Of Operators On Dense Subspaces; 24:15 - Proof;
Lecture 17 Play Video	The Fourier Transform I Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 02:15 - The Fourier Transform; 09:30 - Linearity ; 12:00 - Goal; 15:00 - Properties Of The Fourier Transform; 22:45 - Frequency Vs Fourier Transform;
Lecture 18 Play Video	The Fourier Transform II Lecture with Ole Christensen. Kapitler: 00:00 - Reaching The Goal; 05:00 - Problem With The Fourier Transform; 13:45 - Where Does The Fourier Transform Map Into?; 16:45 - Is F Bounded?; 20:00 - Fourier Transform On L2; 30:00 - Using The Extension Theorem;
Lecture 19 Play Video	Fourier Transform And Wavelets I Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 02:45 - Paley-Wiener Space; 06:30 - The Sinc-Function; 08:30 - Shannon Sampling Theorem; 24:00 - Applications; 33:45 - Convolution;
Lecture 20 Play Video	The Fourier Transform And Wavelets II Lecture with Ole Christensen. Kapitler: 00:00 - Wavelets; 03:00 - Preliminaries; 10:30 - Def.: Wavelet; 23:00 - Multiresolution Analysis; 32:00 - Lemma 8.2.2;
Lecture 21 Play Video	Wavelets And Multiresolution Analysis I Lecture with Ole Christensen. Kapitler: 00:00 - Repetition ; 06:00 - The Key Step (Prop 8.2.6); 29:00 - Construction Of The Wavelet (Thrm 8.2.7); 36:00 - More On The Wavelet (Prop. 8.2.8); 45:00 - Conciderations Concerning Applications;
Lecture 22 Play Video	Wavelets And Multiresolution Analysis II Lecture with Ole Christensen. Kapitler: 00:00 - Status ; 01:00 - How To Construct A Mra; 06:00 - Applications Of Wavelets;
Lecture 23 Play Video	Wavelets And B-Splines I Lecture with Ole Christensen. Kapitler: 00:00 - Repetition: The Construction Of Wavelet Onb; 08:30 - Example: The Haar Mra/Wavelet; 12:30 - More Efficient Compression; 13:30 - Vanishing Moments; 18:00 - Theorem 8.3.3 (Application Of Vanishing Moments); 24:30 - Interpretation Of Thrm 8.3.3; 32:00 - Application Of Wavelets;
Lecture 24 Play Video	Wavelets And B-Splines II Lecture with Ole Christensen. Kapitler: 00:00 - Splines; 05:45 - B-Splines; 10:00 - Properties Of The B-Splines; 24:30 - Proof Of Thrm 10.1.3(I); 44:00 - B-Splines Vs Fourier Transform;
Lecture 25 Play Video	Special Functions And Diff. Equation Course Evaluation Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 07:00 - Legendre Equation; 11:45 - Power Series Method; 15:15 - Theorem 11.2.2; 22:15 - Lemma 11.2.6; 36:00 - Theorem 11.2.7; 66:00 - Evaluation Of The Course;