Part 1 - Introduction and Metric Space

Part 2 - Examples for Metrics

Part 3 - Open and Closed Sets

Part 4 - Sequences, Limits and Closed Sets

Part 5 - Cauchy Sequences and Complete Metric Spaces

Part 6 - Norms and Banach Spaces

Part 7 - Examples of Banach Spaces

Part 8 - Inner Products and Hilbert Spaces

Part 9 - Examples of Inner Products and Hilbert Spaces

Part 10 - Cauchy-Schwarz Inequality

Part 11 - Orthogonality

Part 12 - Continuity

Part 13 - Bounded Operators

Part 14 - Example Operator Norm

Part 15 - Riesz Representation Theorem

Part 16 - Compact Sets

Part 17 - Arzelà–Ascoli Theorem

Part 18 - Compact Operators

Part 19 - Hölder’s Inequality

Part 20 - Minkowski Inequality

Part 21 - Isomorphisms?

Part 22 - Dual Spaces

Part 23 - Dual Space - Example

Part 24 - Uniform Boundedness Principle / Banach–Steinhaus Theorem

Part 25 - Hahn–Banach Theorem

Part 26 - Open Mapping Theorem

Part 27 - Bounded Inverse Theorem and Example

Part 28 - Spectrum of Bounded Operators

Part 29 - Spectrum of Multiplication Operator

Part 30 - Properties of the Spectrum

Part 31 - Spectral Radius

Part 32 - Normal and Self-Adjoint Operators

Part 33 - Spectrum of Compact Operators

Part 34 - Spectral Theorem for Compact Operators