https://github.com/gitcordier/functionalanalysis

Solutions to some exercises from Walter Rudin's Functional Analysis
https://github.com/gitcordier/functionalanalysis

banach-space compactness completeness convex convex-hull distributions exercises-solutions functional-analysis hyperplan linear-functions mathematics measures metric-spaces number-theory tex topology vector-spaces walter-rudin xelatex

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Solutions to some exercises from Walter Rudin's Functional Analysis

• Host: GitHub
• URL: https://github.com/gitcordier/functionalanalysis
• Owner: gitcordier
• License: mit
• Created: 2020-07-24T15:25:07.000Z (almost 5 years ago)
• Default Branch: master
• Last Pushed: 2025-04-04T19:54:18.000Z (24 days ago)
• Last Synced: 2025-04-12T06:58:29.275Z (16 days ago)
• Topics: banach-space, compactness, completeness, convex, convex-hull, distributions, exercises-solutions, functional-analysis, hyperplan, linear-functions, mathematics, measures, metric-spaces, number-theory, tex, topology, vector-spaces, walter-rudin, xelatex
• Language: TeX
• Homepage: https://github.com/gitcordier/FunctionalAnalysis
• Size: 6.59 MB
• Stars: 16
• Watchers: 0
• Forks: 3
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Contributing: CONTRIBUTING.md
  • License: LICENSE

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README

        
# Forewords

A long time ago, I solved a (tiny…) bunch of problems from the Walter Rudin's 

*Functional Analysis*.

I am carefully rewriting my solutions, aiming at the cleanest possible result.

# Requirements

- TeX

- XeLaTeX 

- packages 

  - AMS packages: 

    - amsmath

    - physics 

    - amssymb 

    - mathrsfs 

    - amsthm 

    - mathspec

  - cite

  - geometry

  - xltxtra

  - xunicode

  - xetex

- Optional: Relevant fonts:

    - CMU Serif

    - CMU Typewriter Text Light

You may want to install the latest 

[TexLive](https://www.tug.org)

# Legal issues 

- I do not broadcast nor sell any copy of *Functional Analysis*

- I do not make available any content from this book, 

    **excepted** problems statements 

Feel free to get copies of this great book by yourself. 

*Functional Analysis*' ISBNs are

- ISBN-10: 0070542368

- ISBN-13: 978-0070542365

## Files

- FA_DM.pdf  

  Output from [Xelatex](https://www.tug.org) compilation.  

  You can also get an html output from 

  [Hevea](http://hevea.inria.fr) compilation; see HOWTO.

- FA_DM.tex

- FA_mainmatter.tex

- FA_chapter_1.tex

- chapter_1/ 

  - 1_1.tex

    Basic results that straightforwardly follow from the axioms 

    as given as in section 1.4.

  - 1_2.tex 

    The convex hull of a set A is convex and that is the intersection of 

    all convex set(s) that contain A. 

  - 1_3.tex

  - 1_4.tex

  - 1_5.tex

  - 1_6.tex

  - 1_7.tex  

    I choose to start with this because it is a lovely result, 

    since it connects a topological result 

    (*to be metrizable or not to be*) with number theory.

  - 1_9.tex  

    Continuousness, openness of a linear mapping.

  - 1_10.tex  

    Continuousness, openess of a linear mapping onto a finite dimensional 

    space.Not trivial, since the domain may be infinite-dimensional. 

  - 1_14.tex  

    Alternative ways to the define topology of the test functions space D_K, 

    in the special case K=[0, 1].

  - 1_16.tex   

    This is about showing that a function test topology is independent 

    from the "supremum seminorms" we consider. 

    It is then more than an exercise, it should be regarded as a very part of 

    the textbook (sections 1.44, 1.46).

  - 1_17.tex  

    Given a multi-index $\alpha$, the differential operators $D^\alpha$ is 

    continuous in the test functions topology. 

- chapter_2/ 

  - 2_3/  

      - 2_3.tex  

      In $D_K$, some Lebesgue integrable functions converge to $\delta'$, 

      which is not a Radon measure. 

      Their weak derivatives converge to $\delta''$.

      - 2_3_0_labels.tex  

      References

      - 2_3_0_lemma.tex.  

      Specialization of mean value theorem.

      - 2_3_1_radon_measures.tex  

      Start by looking into $C_0(R)^\ast$.

      - 2_3_2_uniform_bound.tex. 

      $D_K$ topology allows equicontinuity. 

      - 2_3_3_example_1.tex  

      Convergence of Lebesgue integrable functions to $\delta'$ in $D_K$.

      - 2_3_4_example_2.tex  

      Their weak derivatives converge to $\delta''$.  

      - TODO: Add proof that the Dirac derivative is not a Radon measure, 

      to complete the figure.

  - 2_6.tex  

    The Banach-Steinhaus theorem applied to $L^2(T)$ the $L2$ functions of the 

    unit circle of $C$: 

    The series made on the Fourier coefficients may diverge.

    Nevertheless, convergence holds in a dense space 

    (by the the Fejér theorem, for instance). 

    TODO: (?) Add some comparison with the Carleson's theorem. 

  - 2_9.tex  

    Given normed spaces (X, Y, Z), any continuous bilinear mapping 

    $B: (x, y) \in X\times Y \mapsto B(x, y) \in Z$ is bounded. 

    Thoses spaces need not be complete. An easy example is given by 

    $B(f, g)= fg$ where $f$ and $g$ keep in $C_c(R)$. 

  - 2_10.tex  

    A bilinear mapping that is continuous at the origin is continuous. 

    Actually, 2.09 contains all the relevant material. 

    In the more general topological vector space context proof, 

    the norm is replaced by Minkowski functionals on balanced open sets.

  - 2_12.tex  

    A bilinear mapping that is separately continuous, but not continuous.

  - 2_15.tex  

    In a F-space X, the complement C of a subgroup Y is not 

    of the first category, unless X=Y.

    To sum it up: If Y is a proper subspace, 

    then Y is of the first category BUT its complement C 

    is of the second category, as X is. The proof is flawed/not achevied. I have written a correct proof that I will push soon. 

  - 2_16.tex  

    A simpler version of the closed graph theorem. 

    Roughly speaking, compactness replaces completeness. Compactness cannot 

    be dropped: A counterexample is given.

- chapter 3/

  - 3_4.tex

  - 3_11.tex  

    Meagerness of the polar (in the infinite dimensional case) 

    of the neighborhoods of the origin: 

    Hahn-Banach theorem and polar. We only involve the weak star 

    -closedness of the polar, not its weak star-compactness!

- chapter 4/

  - 4_1.tex

  - 4_13.tex

  - 4_15.tex

- chapter 6/

  - 6_1.tex

  - 6_9.tex

  - 6_9.tex

- LICENSE

- HOWTO