CAAM 540 · Applied Functional Analysis
Fall 2008 · Rice University
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Notes and Supplemental Material
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| Lecture 12: |
The dual space, reflexive spaces, intro to linear operators
Reading: Young, Chapter 6, 7.
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| Lecture 11: |
Linear functionals, BLT Theorem, Riesz Representation Theorem
Reading: Young, Chapter 6.
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| Lecture 10: |
Stone-Weierstrass Theorem
Reading: Yosida, Functional Analysis, pages 6-11.
Bishop's Theorem generalizes Stone-Weierstrass; see Rudin, Functional Analysis, 2nd ed, pages 121-124.
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| Lecture 9: |
Completeness of L2
Reading: Young, Chapter 5; Körner, Fourier Analysis, pages 67-75.
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| Lecture 8: |
Orthogonal complements; completeness of L2
Reading: Young, Chapter 5.
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| Lecture 7: |
Complete orthonormal sequences; separable Hilbert spaces
A sequence is a c.o.n.s iff its closed linear span is the whole space iff Parseval's identity holds.
See A Basis Theory Primer by Christopher Heil
The Chebfun project
Reading: Young, Chapter 4.
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| Lecture 6: |
General Fourier Series
Reading: Young, Chapter 4.
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| Lecture 5: |
Best Approximation in Hilbert Space
Any element of a Hilbert space has a unique
best approximation from a closed, convex subset.
One counterexample was taken from the book Functional Analysis by Peter Lax
The example of best approximation in finite elements was adapted from
the basic error analysis in Brenner and Scott
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| Lecture 4: |
Banach and Hilbert spaces
Reading: Young, Chapter 3.
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| Lecture 3: |
Topological properties of normed spaces
In finite dimensional spaces,
all norms generate the same topology.
Reading: Young, Chapter 2.
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| Lecture 2: |
Normed linear spaces; topology
Reading: Young, Chapter 2.
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| Lecture 1: |
Introduction; inner product spaces
Reading: Young, Chapter 1.
Some supplemental resources:
Haïm Brezis, Analyse Fonctionelle, Dunod, 1983
Tosio Kato, Perturbation Theory for Linear Operators, Springer, 1980
Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978
Peter D. Lax, Functional Analysis, Wiley, 2002
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I:
Functional Analysis, Academic Press, 1980
Walter Rudin, Functional Analysis 2nd ed., 1991
Kôsaku Yosida, Functional Analysis, Springer, 1980
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