# Real Analysis: Math 209

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REAL ANALYSIS: MATH 209

MATH 209A

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material: • Preliminaries — Chapter 0 • Measures — Chapter 1 • Integration — Chapter 2

Topics include: • Properties of both abstract and Lebesgue-Stieltjes measures • Caratheodory extension process constructing a measure on a sigma-algebra from a premeasure on an algebra; construction of Lebesgue-Stieltjes measure via this process • Borel measures; complete measures; sigma-ﬁnite measure spaces • Properties of measurable functions • Abstract integration as well as Lebesgue integration on Rn • Dominated and monotone convergence theorems, Fatou’s Lemma • Special examples: Cantor sets, Cantor function, construction of a non-Lebesgue measurable subset of [0, 1]. • Modes of convergence: pointwise, uniform, almost everywhere, in measure, in L1-norm, and implications between modes of convergence; Egoroﬀ’s and Lusin’s theorems • Product measures: Fubini’s theorem and Tonelli’s theorem • Relation of Lebesgue integral to Riemann integral

MATH 209B

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material:

• Signed Measures and Diﬀerentiation — Chapter 3 • Point Set Topology — Sections 4.1—4.7 • Normed Vector Spaces, Linear Functionals, and the Baire Category Theorem and

its Consequences — Sections 5.1—5.3 • Topological vector spaces—Chapter 5.4

Topics include: • Radon-Nikodym theorem; Hahn, Jordan, and Lebesgue decompositions • Lebesgue’s diﬀerentiation theorem in Rn; functions of bounded variation, absolute continuity • Nets, Urysohn’s lemma, compactness, the Stone–Weierstrass theorem, product topologies, Tychonoﬀ’s theorem • Normed vector spaces: Banach spaces, quotients, adjoints, Hahn-Banach Theorem, Baire category theorem, open mapping theorem, closed graph theorem, the uniform boundedness principle • Topological vector spaces: weak topology, weak-∗ topology, Alaoglu’s theorem

MATH 209C

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material:

• Hilbert spaces — Section 5.5 • Lp spaces — Chapter 6 • The dual of Cc(X) and C0(X) — Sections 7.1 and 7.3 • Fourier analysis — Chapter 8.1—8.3 and 8.7 • Distributions — Chapters 9.1 and 9.2

Topics include:

• Hilbert spaces: Cauchy-Schwarz inequality, parallelogram law, Pythagorean theorem, Bessel’s inequality, Parseval’s identity, Riesz representation theorem, orthonormal bases

• Lp and lp spaces: H¨older and Minkowski inequalities, duals of these spaces

• Various classes of functions: C∞ , Cc∞, Cc , C0 and their duals • Fourier analysis on Tn and Rn, convolution, Fourier inversion theorem, Young’s

and Hausdorﬀ-Young inequalities, applications to partial diﬀerential equations

• Schwarz functions and tempered distributions, convolution of tempered distributions, the Fourier transform of tempered distributions

MATH 209A

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material: • Preliminaries — Chapter 0 • Measures — Chapter 1 • Integration — Chapter 2

Topics include: • Properties of both abstract and Lebesgue-Stieltjes measures • Caratheodory extension process constructing a measure on a sigma-algebra from a premeasure on an algebra; construction of Lebesgue-Stieltjes measure via this process • Borel measures; complete measures; sigma-ﬁnite measure spaces • Properties of measurable functions • Abstract integration as well as Lebesgue integration on Rn • Dominated and monotone convergence theorems, Fatou’s Lemma • Special examples: Cantor sets, Cantor function, construction of a non-Lebesgue measurable subset of [0, 1]. • Modes of convergence: pointwise, uniform, almost everywhere, in measure, in L1-norm, and implications between modes of convergence; Egoroﬀ’s and Lusin’s theorems • Product measures: Fubini’s theorem and Tonelli’s theorem • Relation of Lebesgue integral to Riemann integral

MATH 209B

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material:

• Signed Measures and Diﬀerentiation — Chapter 3 • Point Set Topology — Sections 4.1—4.7 • Normed Vector Spaces, Linear Functionals, and the Baire Category Theorem and

its Consequences — Sections 5.1—5.3 • Topological vector spaces—Chapter 5.4

Topics include: • Radon-Nikodym theorem; Hahn, Jordan, and Lebesgue decompositions • Lebesgue’s diﬀerentiation theorem in Rn; functions of bounded variation, absolute continuity • Nets, Urysohn’s lemma, compactness, the Stone–Weierstrass theorem, product topologies, Tychonoﬀ’s theorem • Normed vector spaces: Banach spaces, quotients, adjoints, Hahn-Banach Theorem, Baire category theorem, open mapping theorem, closed graph theorem, the uniform boundedness principle • Topological vector spaces: weak topology, weak-∗ topology, Alaoglu’s theorem

MATH 209C

Textbook. The textbook is Gerald Folland’s Real Analysis.

Reference. A very useful reference is H. L. Royden’s Real Analysis, or the 4th edition of this book written by Royden and P. Fitzpatrick.

We will cover approximately the following material:

• Hilbert spaces — Section 5.5 • Lp spaces — Chapter 6 • The dual of Cc(X) and C0(X) — Sections 7.1 and 7.3 • Fourier analysis — Chapter 8.1—8.3 and 8.7 • Distributions — Chapters 9.1 and 9.2

Topics include:

• Hilbert spaces: Cauchy-Schwarz inequality, parallelogram law, Pythagorean theorem, Bessel’s inequality, Parseval’s identity, Riesz representation theorem, orthonormal bases

• Lp and lp spaces: H¨older and Minkowski inequalities, duals of these spaces

• Various classes of functions: C∞ , Cc∞, Cc , C0 and their duals • Fourier analysis on Tn and Rn, convolution, Fourier inversion theorem, Young’s

and Hausdorﬀ-Young inequalities, applications to partial diﬀerential equations

• Schwarz functions and tempered distributions, convolution of tempered distributions, the Fourier transform of tempered distributions

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