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
• LuaLaTeX
• Packages
  • fontspec
  • xparse
  • AMS packages:
    • amsmath
    • physics
    • amssymb
    • mathrsfs
    • amsthm
    • mathspec
  • cite
  • geometry
  • caption
• Optional: Relevant fonts:
  • New Computer Modern 10 (NewCM10)

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, except 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 compilation.You can also get an html output from Hevea compilation, see HOWTO.
• FA_DM.tex, the master document.
• FA_mainmatter.tex
• FA_chapter_1.tex
• chapter_1/
  • 1_1.tex Basic results that straightforwardly follow from the axioms as given 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_07.tex This is a lovely result, since it connects a topological result (to be metrizable or not to be) with number theory.
  • 1_9.tex Continuity, openness of a linear mapping.
  • 1_10.tex Continuity, openness of a linear mapping onto a finite dimensional space. Not trivial, since the domain may be infinite-dimensional.
  • 1_14.tex Alternative ways to define the 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 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_03.tex In $D_K$, some Lebesgue integrable functions converge to $\delta'$, which is not a Radon measure. Their weak derivatives converge to $\delta''$.
  • 2_09.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 jointly 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
• annex/
  • annex_vector_spaces.tex Lemma for (j) of Exercise 1.1 (1_01.tex)
  • annex_mean_value.tex Examples and Lemma for Exercise 2.3 (2.03.tex).
  • annex_Dirac.tex
• LICENSE
• HOWTO