User:Lord Farin/Long-Term Projects/Conway Course

Processing of 'A Course in Functional Analysis'

1990: John B. Conway: A Course in Functional Analysis (2nd ed.)

1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Example $1.7$

1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra: Proposition $1.1$

Progress thus far

Celebration: Chapter $I$ covered! Lord_Farin 08:05, 16 January 2012 (EST)

Up to $II$ / p.26. Lord_Farin 04:00, 18 January 2012 (EST)

Up to $II.1.2$ / p.27. Lord_Farin 18:34, 23 January 2012 (EST)

Up to $II.2$ / p.31. Lord_Farin 08:55, 24 January 2012 (EST)

Up to $II.2.6$ / p.32. Lord_Farin 08:34, 27 January 2012 (EST)

Up to $II.2.7$ / p.32. Lord_Farin 08:36, 28 January 2012 (EST)

Up to $II.2.11$ / p.33. Lord_Farin 11:26, 30 January 2012 (EST)

Up to $II.2.17$ / p.35. Lord_Farin 05:09, 31 January 2012 (EST)

Up to $II.3$ / p.36. Lord_Farin 10:54, 31 January 2012 (EST)

Up to $II.3.3$ / p.37. Lord_Farin 10:35, 1 February 2012 (EST)

Up to $II.3.5$ / p.38. Lord_Farin 18:12, 2 February 2012 (EST)

Up to $II.3 \text{ Exercises}$ / p.40. Lord_Farin 10:42, 3 February 2012 (EST)

Up to $II.4$ / p.41. Lord_Farin 15:39, 14 February 2012 (EST)

Up to $II.4.6$ / p.42. Lord_Farin 09:18, 24 February 2012 (EST)

Up to $II.5$ / p.46. Lord_Farin 17:59, 27 February 2012 (EST)

Up to $II.7.6$ / p.55. Lord_Farin 08:50, 2 March 2012 (EST)

Reference fixed to Second Edition. Next step will be to fix presentation of the citations, and complete the proofs where missing. — Lord_Farin (talk) 16:47, 22 September 2022 (UTC)

Up to $I.1.7$. Quite more work this time around due to increased rigour all around. — Lord_Farin (talk) 18:10, 12 October 2022 (UTC)

Up to $I.2.6$. Some nice improvements of the flow. — Lord_Farin (talk) 19:08, 14 October 2022 (UTC)

Finished $I.2$. — Lord_Farin (talk) 07:16, 25 October 2022 (UTC)

Finished $I.3$. — Lord_Farin (talk) 07:04, 29 October 2022 (UTC)

Start covering the Appendix A at Definition:Linearly Independent Set to create/verify solid foundations on nets. — Lord_Farin (talk) 16:36, 31 October 2022 (UTC)

Finished $A.1$. — Lord_Farin (talk) 13:56, 4 November 2022 (UTC)

Refactorings due

• Strict analysis of where complete subfields of $\C$ are required, plus supplemental material that a complete subfield of $\C$ is either $\R$ or $\C$.
• Definition:Linear Subspace vs Definition:Closed Linear Subspace, in the flow after Definition:Orthogonal Projection
• The categories around linear functionals/transformations are a complete mess
• Separate categories and possibly pages for $L^2$ spaces

Skipped thus far (that is, what needs to be done still)

• $I.1.8$ Absolutely continuous complex functions
• $I.1.10-13: $ Bergman Spaces $L^2_a(G)$
• $I.2.10$ Proof of Double Orthocomplement is Closed Linear Span, which needs some of the exercises
• $I.3$ Exercises 3,4 about power series (after Riesz Representation Theorem (Hilbert Spaces)/Examples/Space of Square Summable Mappings)
• $I.3$ Exercise 5 about $I.1.8$ (see above)
• $I.3$ Exercise 6 evaluation of derivative at point is not bounded
• $I.4.4-5$ Proofs of examples of Basis (Hilbert Space)/Examples
• $I.4.6$ Gram-Schmidt Orthogonalization (easy proof)
• $I.4.7$ Orthogonal Projection onto Closed Linear Span, Conway gives a simpler proof
• $I.5.6-11:$ Rigorous foundations of Fourier analysis and $L^2_\C[0,2\pi]$.
• $II.1.5:$ Bounded functions induce bounded linear operators on $L^2(\mu)$.
• $II.1.6:$ Integral operators with certain kernels are bounded linear operators.
• Maybe some more of $II.1 \text{ Exercises}, II.2 \text{ Exercises}$.
• $II.2.8,9:$ Examples which are useful but their basis hasn't been covered.
• $II.2.10:$ Unilateral shift on $\ell^2(\N)$
• $II.4.7-8$: Integral operators with certain kernels are compact operators
• $II.6$: Sturm-Liouville theory