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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Functional Analysis
( Uncountable  Dimensionality )


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Théorie des opérations linéaires  (Banach spaces)  by  Stefan Banach  (1932).
Functional Analysis  by  Douglas N. Arnold  (PSU, Math 503, Spring 1997).
Functional Analysis Notes  by  Fengbo Hang  (Spring 2009)
Functional Analysis  (1973, 1991)   by  Walter Rudin (1921-2010)
Hahn-Banach Theorems   by  Sudhir Hanmantrao Kulkarni.
Wikipedia :   Functional analysis  |  Linear transformation  |  Clifford Algebra
 Jacques Hadamard 
Jacques Hadamard

Functional Analysis

functional  (or  form)  is a mapping that assigns a scalar value to a function.  One example of a linear functional is the integral over  [a,b]  of an integrable scalar functions of a real variable.  The word "functional" itself was introduced in that sense by  Jacques Hadamard  in 1910.
The concept of a function of functions goes back to the early days of Lagrange's  calculus of variation (1744) before it was called that (1766).
Functions over any infinite set entail a  linear space  of functionals whose dimensionality is uncountable.  This forms the main backdrop of what's now called  functional analysis.
   Paul Levy 
 1886-1971; X1904
Paul Lévy  1886-1971
Paul Lévy (1886-1971; X1904) wrote his doctoral dissertation on functional analysis (1911) under Jacques Hadamard and Vito Volterra (1860-1940)  in the spirit pioneered by Volterra and Salvatore Pincherle (1853-1936):  They considered functions whose arguments are curves, surfaces or sequences.  Lévy wrote an early textbook on the subject,  based on his own lectures at the  Collège de France,  entitled  " Leçons d'analyse fonctionnelle "  (1922).  In it,  Lévy could popularize concepts due to  René Gateaux  (1889-1914)  including  Gateaux integrals,  which might otherwise have been lost  (Lévy had been entrusted in 1919 with the unpublished notes of Gateaux, left at his mother's home when he was drafted to serve as a lieutenant in the first days of WWI.)
From 1920 to 1959,  Paul Lévy  held the chair of analysis at Polytechnique,  which  Lagrange  had inaugurated in 1794.  Lévy had only four doctoral students  (including Benoît Mandelbrot, of fractal fame).  His successor was his own son-in-law, the legendary inventor of the theory of distributions (Nov. 1944):  Laurent Schwartz (1915-2002).  Schwartz held the position from 1959 to 1980 and, incidentally, taught  me  functional analysis  (mostly Hilbertian)  in the Fall of 1977, starting with the  fundamental theorem of functional analysis,  the  Hahn-Banach theorem...

(2012-09-25)   Trailblazing, 100 years ago...  (Eduard Helly, 1912)
Hahn-Banach theorem.  15 years before Hahn, 17 years before Banach.

In 1912, the Austrian mathematician  Eduard Helly  (1884-1943) focused on the space  C[a,b]  of the continuous real functions of a real variable over the closed interval  [a,b].  The  dual  of that space is denoted  C[a,b]*  and consists of all the  continuous  linear functionals  F  assigning a real value  to every function  u  belonging to  C[a,b].

F :  C[a,b]    ®    R
u   butt®   F(u)

Every such functional can be uniquely expressed as a  Stieltjes integral :

( "u Î C[a,b] )       F(u)   =    ò  b   u(x) dm(x)

In fact,  this equivalence is one way  Stieltjes measures  could be introduced.  Those measures are arguably the forerunners of the general  distributions  Laurent Schwartz (1915-2002) would devise in 1944  (actually, Stieltjes measures are just a special type of distributions).

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Helly gave essentially the same proof as what Hans Hahn would publish for the general case 15 years later.

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 still working on this one...

Stieltjes integration   |   Eduard Helly (1884-1943)

(2012-09-22)   Riesz extension theorem (Marcel Riesz, 1923)

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Hahn-Banach Theorem   |   Hans Hahn (1879-1934)   |   Stefan Banach (1892-1945)
Riesz extension theorem (1923)   |   Marcel Riesz (1886-1969)

(2012-09-25)   Sublinear functionals  (or  Banach functionals )
Functionals that are  positively homogeneous  and  subadditive.

Norms or seminorms are the most common examples of sublinear functionals.  However, just  positive homogeneity  and  subadditivity  are sufficient to establish the fundamental  Hahn-Banach theorem.

A functional over a real vector space  p  is said to be  sublinear  when:

  • It's  subadditive :   p (x+y)   ≤   p(x) + p(y)
  • It's  positively homogeneous  :   " k > 0 ,   p (k x)  =  k p(x).

Clearly, a sublinear functional is also  convex,  which is to say:

" t Î [0,1]       p ( t x  +  (1-t)y )   ≤   t p(x)  +  (1-t) p(y)

Surprisingly enough, it's  not necessary  to postulate that the values of  p  are nonnegative.

Caution sign
In  computer science, the qualifer  sublinear  applies to a real function of a real variable which is  negligible  in the neighborhood of infinity, compared to  any  linear function.  To a computer scientist, a linear function is not sublinear!  Use the term  "Banach functional"  when there's a risk of confusing the above with computer jargon...

Banach functional   |   Minkowski functional   |   convex function

(2012-09-21)  Dominated Hahn-Banach extension theorem  (Hahn, 1927)
Any linear (continuous) functional on a subspace  F  of a linear space  E  has a norm-preserving (continuous) linear extension to the whole space.

So stated, the theorem applies to  normed spaces,  but the result is more general, as it applies to functions dominated by any  Banach functional,  which may or may not be equal to the norm itself.  The general theorem is:

Theorem :   A linear functional defined on a subspace and dominated by some sublinear functional  p  can be extended to a linear function dominated by that same  p  over the whole space.

This key result, obtained by Hans Hahn in 1927 for real linear spaces, is now called the  (analytic)  Hahn-Banach theorem  together with its  (geometric)  equivalent counterpart  due to S. Banach (1929).  The unified name was introduced by by H.F. Bohnenblust and Andrew Sobcyzk, in 1938, as they credited F. Murray for a general way (1936) of extending the analytic result to  complex spaces, in the wake of the publication by S. Banach of the first textbook on the topic  (1932).

If  E  is  separable, the theorem can be proved without invoking any type of axiomatic choice  (like Zorn's lemma)  but the unrestricted result does require that.  The full force of the  axiom of choice  isn't needed but some weaker form of axiomatic choice is.  Following H. Hahn (1927) and S. Banach (1929) we'll first consider only the case of  real  vector spaces:

Proof of the special case when  F  is an hyperplane of   E :

E   =   F Å G     where  G  is one-dimensional.

Let's choose any nonzero element  g  in  G  and express uniquely any vector in  E  as a sum of a vector x in  E  and a multiple of  g :

x  +  t g

A linear extension  û  of a linear functional  u  on  F  is fully determined by just one real constant  a = û(g)  which we'll use as an identifying subscript:

" x Î F" t Î R,   ûa ( x  +  t g )   =   u(x) + t a

If we know that  u  is dominated by the  sublinear functional  p  over  F,  we seek the conditions for  ûa  to be dominated by  p  over  E,  namely:

" x Î F" t Î R,     u(x) + t a   ≤   p ( x  +  t g )

If  t  is zero,  this is trivially satisfied for any a  (because  u  is dominated by  p  over  F).  Otherwise, we separate the cases where  t = +k  and  t = -k  for some  positive  k,  so we can use the positive homogeneity of  p.  Putting respectively  y = x/k,   and  z = -x/k,  the above is thus equivalent to:

" y Î F,    u(y) + a   ≤   p ( y  +  g )
" z Î F,    u(z) - a   ≤   p ( z  -  g )

That pair of statements can be rewritten as:

Sup zÎF { u(z) - p (z - g) }   ≤   a   ≤   Inf yÎF { p (y + g) - u(y) }

A nonempty range of acceptable values of  a  is so described, since:

" z Î F" y Î F,      u(z) - p (z - g)   ≤   p (y + g) - u(y)
or, equivalently     u(z) + u(y)   ≤   p (z - g) + p (y + g)

Which is true because we have   u(z) + u(y)   =   u(z+y)   and, moreover:

u(z+y)   ≤   p(z+y)   =   p(z - g + y + g)   ≤   p (z - g) + p (y + g)     QED (Halmos symbol)

Proof of the general case, using Zorn's lemma :

Let  p  be a  sublinear form  over the real vector space  E.  Let  u  be a linear form defined over the subspace  F  and dominated by  p :

" x Î F,   u(x)  ≤  p(x)

A dominated linear extension of  u  over a larger subspace  G  is determined by the ordered pair  (û,G)  where  û  is a linear form defined on  G  such that:

" x Î F,   û(x)   =   u(x)
" y Î G,   û(y)   ≤   p(y)

We may define a partial ordering relation among all such pairs as follows:

{ (û,G)   ≤   (û',G' ) }   Û   { G Í G'" x Î G,  û(x) = û'(x) }

By Zorn's lemma, there's a maximal element  (â,A)  for this ordering.

If  A  was a proper subspace of  E,  there would be a vector  g  outside of  A  or, equivalently, a subspace  A'  of which  A  would be an hyperplane.  By the special case already proven, we could extend  â  to  A', which would contradict the maximality of  (â,A).  Therefore,  A = E  and  â  is indeed an extension of  u  dominated by  p  and defined over the entire space  EQED (Halmos symbol)

General proofs using weaker forms of  Choice :

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Proof for restricted types of vector spaces, without  Choice :

For some types of vector spaces, the Hahn-Banach theorem can be given a  constructive  proof  (strictly within Zermelo-Fraenkel set theory, without invoking the  Axiom of choice or any weaker substitute).  This includes:

Arguably, the simplest Banach space outside of the above classes is the  limit  Lebesgue sequence space   l_p ¥  whose standard norm is given by:

|| x ||¥   =   || (x1 , x2 , x3 , ...) ||¥   =   Supn |xn |

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Hahn-Banach Theorem   |   Hans Hahn (1879-1934)   |   Stefan Banach (1892-1945)
The Hahn-Banach Theorem  by  Gabriel Nagy   |   The Hahn-Banach Theorem  by  Ambar N. Sengupta
Hahn-Banach extension theorems and existence of linear functionals  by  Lawrence Baggett
Hahn-Banach Theorem for Real Vector Spaces  (video)  by  P.D. Srivastava.
Duality and the Hahn-Banach theorem  by  Terence Tao   (2009-01-26)
The Hahn-Banach theorem, Menger's theorem and Helly's theorem  by  Terence Tao   (2007-11-30)
Hahn-Banach without Choice :  Forum discussion (2010).
ZF implies a weak version of Hahn-Banach :  Forum discussion (2013).
The Hahn-Banach Theorem: The Life and Times  by  Lawrence Narici  and  Edward Beckenstein  (2002).

(2012-09-21)   Hahn-Banach separation theorem (S. Banach, 1929)
Banach  gave a new setting to the theorem proven by Hahn in 1927.

Hyperplane Separation Theorem :

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Hahn-Banach Theorems  by  Sudhir Hanmantrao Kulkarni
Hyperplane separation theorem   }   Hermann Minkowski (1864-1909)
MIT 18.409, by Jonathan Kelner :   Convex geometry | Separating hyperplanes
Direct proof of the separation theorem of Hahn-Banach

(2012-09-21)   Hahn-Bahnach for  complex  or  quaternionic  spaces :
The Hahn-Banach theorem generalizes to  hypercomplex  linear spaces.

The Hahn-Banach extension theorem was generalized to complex linear spaces by  Francis J. Murray (1911-1996)  in his doctoral dissertation  (1936).  Murray was only concerned with the Lebesgue space  L[a,b]  (p > 1)  of the complex functions of a bounded real variable.  However, his methods were readily applicable to any other complex linear space.

Bohnenblust & Sobcyzk (1938)  acknowledged that, gave the Hahn-Banach theorem its final name and popularized its complex version.  (They also showed that a linear extension isn't always possible for a complex-valued functional defined on a  real  subspace not stable under complex scaling.)

The key is to realize that any complex linear functional  f  can be expressed purely in terms of its  real part  (an easy exercise left to the reader)  namely:

f (x)   =   Re ( f (x) )  -  i Re ( f (ix) )

Thus, a complex-linear form on a complex subspace can be extended just like its real part can  (using the Hahn-Banach theorem for real spaces).

For a quaternionic-linear form  f  in quaternionic space, the key relation is:

f (x)   =   Re ( f (x) )  -  i Re ( f (ix) )  -  j Re ( f (jx) )  -  k Re ( f (kx) )

In other words, there is a  real-valued  linear functional  h  such that:

f (x)   =   h(x)  -  i h(ix)  -  j h(jx)  -  k h(kx)

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Kudos:  The aforementioned 1938 paper of Bohnenblust & Sobcyzk is the birth certificate of our "fundamental theorem of functional analysis",  since that's where it appeared under the name of "Hahn-Banach" for the first time.  At the time, Henri Frédéric Bohnenblust {1906-2000) was actually the thesis advisor of his co-author Andrew F. Sobczyk (1915-1981).  "Boni" was a Swiss-born American mathematician who graduated from the  ETH Zürich in 1928 and obtained his doctorate from Princeton in 1931.  He would later make the cover of Time magazine (May 6, 1966) with 9 other "great college teachers".
    Paolo G. Comba (1926-) is another former doctoral student of Boni's and an amateur astronomer who built Prescott Observatory when he retired in 1991.  He discovered 654 asteroids.  On 1997-12-27, Paul Comba discovered a minor planet which he decided to name in honor of Boni:  15938 Bohnenblust  (1997 YA8).  The naming of the Hahn-Banach theorem is another more arcane part of Boni's legacy.

Hahn-Banach Theorem for Complex Vector Spaces  (video)  by  P.D. Srivastava.
"On extensions of linear functions in complex and quaternionic linear spaces",
by G. A. Suhomlinov (1938) Mat. Sbornik 3, 353-358.
"Extensions of functionals on octonionic linear spaces",
by J.L. Lewis (1988) Acta Math. Hung., 52 (3-4) 249-253

(2012-09-24)   The two Baire Category Theorems (1899)
1.   Every [separable] complete space is a Baire space.
2.   Every [separable] locally compact Hausdorff space is a Baire space.

By definition, a  Baire space  is a topological space where any countable intersection of open dense sets is dense.

The first statement  (not restricted to separable spaces)  is equivalent to the  axiom of dependent choice  (it's more popular in the form:  "A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets").

Both statements can be proven in pure ZF set theory  (without any choice principle)  if restricted to  separable spaces.

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Baire category theorem (1899)   |   Baire property (almost open sets)   |   René-Louis Baire (1814-1932)

(2012-09-24)   Uniform Boundedness Principle  (Banach-Steinhaus)
First published by Stefan Banach and Hugo Steinhaus in 1927).

The Banach-Steinhaus theorem was proven independently by Hans Hahn.

Consider a Banach space  X,  a normed space  Y  and a set  F  of continuous linear operators from  X  to  Y.  The  Banach-Steinhaus theorem  says that if the operators of  F  are bounded at every point of  X, they are  uniformly  bounded over  X.

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Uniform Boundedness Principle  (Banach-Steinhaus theorem)   |   Video : The Uniform Boundedness Principle  by Joel Feinstein  (University of Nottingham)

(2012-09-24)   Open Mapping Theorem

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Open mapping theorem  (Banach-Schauder theorem)   |   Julius Schauder (1899-1943)

(2012-09-24)   Closed Graph Theorem
A consequence of the Open Mapping Theorem.

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Closed graph theorem

(2012-09-24)   Bounded Inverse Theorem
A consequence of the Open Mapping Theorem.

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Bounded inverse theorem

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