Dive into a comprehensive 4.5-hour video series on functional analysis, exploring key concepts from metric spaces to spectral theory. Begin with the fundamentals of metric spaces, open and closed sets, and sequences. Progress through norms, Banach spaces, and Hilbert spaces, examining important theorems like Cauchy-Schwarz Inequality and Riesz Representation. Investigate continuity, bounded operators, and compact sets, including the Arzelà –Ascoli theorem. Delve into advanced topics such as Hölder's and Minkowski inequalities, dual spaces, and pivotal principles like the Uniform Boundedness Principle and Hahn-Banach theorem. Conclude with an in-depth exploration of spectral theory, covering the spectrum of bounded operators, multiplication operators, and properties of normal and self-adjoint operators.
Overview
Syllabus
Functional Analysis - Part 1 - Metric Space.
Functional Analysis - Part 2 - Examples for metrics.
Functional Analysis - Part 3 - Open and closed sets.
Functional Analysis - Part 4 - Sequences, limits and closed sets.
Functional Analysis - Part 5 - Cauchy sequences and complete metric spaces.
Functional Analysis - Part 6 - Norms and Banach spaces.
Functional Analysis - Part 7 - Examples of Banach spaces.
Functional Analysis - Part 8 - Inner Products and Hilbert Spaces.
Functional Analysis - Part 9 - Examples of Inner Products and Hilbert Spaces.
Functional Analysis - Part 10 - Cauchy-Schwarz Inequality.
Functional Analysis - Part 11 - Orthogonality.
Functional Analysis - Part 12 - Continuity.
Functional Analysis - Part 13 - Bounded Operators.
Functional Analysis - Part 14 - Example Operator Norm.
Functional Analysis - Part 15 - Riesz Representation Theorem.
Functional Analysis - Part 16 - Compact Sets.
Functional Analysis - Part 17 - Arzelà–Ascoli theorem.
Functional Analysis - Part 18 - Compact Operators.
Functional Analysis - Part 19 - Hölder's Inequality.
Functional Analysis - Part 20 - Minkowski inequality.
Functional Analysis - Part 21 - Isomorphisms?.
Functional Analysis - Part 22 - Dual spaces.
Functional Analysis - Part 23 - Dual space - Example.
Functional Analysis - Part 24 - Uniform Boundedness Principle / Banach–Steinhaus Theorem.
Functional Analysis - Part 25 - Hahn–Banach theorem.
Functional Analysis - Part 26 - Open Mapping Theorem.
Functional Analysis - Part 27 - Bounded Inverse Theorem and Example.
Spectral Theory 1 - Spectrum of Bounded Operators (Functional Analysis - Part 28).
Spectral Theory 2 - Spectrum of Multiplication Operator (Functional Analysis - Part 29).
Spectral Theory 3 - Properties of the spectrum (Functional Analysis - Part 30).
Spectral Theory 4 - Spectral Radius (Functional Analysis - Part 31).
Spectral Theory 5 - Normal and Self-Adjoint Operators (Functional Analysis - Part 32).
Taught by
The Bright Side of Mathematics
