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

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 compilation.
  You can also get an html output from Hevea 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.
  • 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