Introduction to Functional Analysis

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Lecture 1: Basic Banach Space Theory

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Lecture 2: Bounded Linear Operators

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Lecture 3: Quotient Spaces, the Baire Category Theorem and the Uniform Boundedness Theorem

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Lecture 4: The Open Mapping Theorem and the Closed Graph Theorem

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Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem

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Lecture 6: The Double Dual and the Outer Measure of a Subset of Real Numbers

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Lecture 7: Sigma Algebras

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Lecture 8: Lebesgue Measurable Subsets and Measure

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Lecture 9: Lebesgue Measurable Functions

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Lecture 10: Simple Functions

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Lecture 11: The Lebesgue Integral of a Nonnegative Function and Convergence Theorems

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Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence...

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Lecture 13: Lp Space Theory

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Lecture 14: Basic Hilbert Space Theory

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Lecture 15: Orthonormal Bases and Fourier Series

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Lecture 16: Fejer’s Theorem and Convergence of Fourier Series

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Lecture 17: Minimizers, Orthogonal Complements and the Riesz Representation Theorem

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Lecture 18: The Adjoint of a Bounded Linear Operator on a Hilbert Space

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Lecture 19: Compact Subsets of a Hilbert Space and Finite-Rank Operators

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Lecture 20: Compact Operators and the Spectrum of a Bounded Linear Operator on a Hilbert Space

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Lecture 21: The Spectrum of Self-Adjoint Operators and the Eigenspaces of Compact Self-Adjoint...

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Lecture 22: The Spectral Theorem for a Compact Self-Adjoint Operator

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Lecture 23: The Dirichlet Problem on an Interval