Undergraduate Analysis
- References
- Rudin: Chapters 2, 3, 4, 5, 7
- Folland: Section 0.6
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Continuity and differentiation of functions \(f: {\mathbf{R}}\to {\mathbf{R}}\)
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Metric spaces
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Compactness in analysis
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Sequences and series
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Uniform convergence
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Uniform continuity
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Taylor’s theorem
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Weierstrass Approximation Theorem
Measure and Integration
- References:
- Folland, Chapters 1, 2
- Stein and Shakarchi, Chapters 1, 2, 6
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Measures on \({\mathbf{R}}^n\) and on \(\sigma{\hbox{-}}\)algebras
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Measurable functions
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Integrable functions
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Convergence theorems:
- Fatou’s Lemma
- The Monotone Convergence Theorem
- The Dominated Convergence Theorem
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Egorov’s theorem
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Notions of convergence:
- Uniform
- Pointwise
- Almost everywhere
- In norm
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Fubini and Tonelli theorems
Function Spaces
- References:
- Folland: Sections 5.2, 5.5, 6.2;
- Stein and Shakarchi: Chapter 4.
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The Banach spaces \(L^1\) and \(L^\infty\):
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Completeness
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Convolutions
- Approximations to the identity
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Linear functionals and
- \(L^\infty\) as the dual of \(L^1\)
Hilbert space and \(L^2\) spaces:
- References:
- Folland: Sections 5.2, 5.5, 6.2;
- Stein and Shakarchi: Chapter 4.
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Schwarz inequality
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Orthogonality
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Linear functionals
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The Riesz representation theorem
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Bessel’s inequality
- Orthonormal bases
- Parseval’s identity
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Trigonometric series:
- Trigonometric polynomials are dense in \(( C([0, 1]), {\left\lVert {{-}} \right\rVert}_\infty)\)
- Trigonometric polynomials are dense in \(( L^2([0, 1]), {\left\lVert {{-}} \right\rVert}_2 )\)