A functional
is a mapping which 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).
The familiar finite-dimensional
ordinary vector spaceE can be construed
as the set of linear functions over a finite set I (consisting of finitely many
base vectors). The standard
topology on E
is the product topology induced by the topology of its scalar field K and that makes all
linear functionals continuous. Those linear functionals
form a vector space E* of the same dimension, known as the dual space
(or space of covectors).
On the other hand, functions over an infinite set I
form a vector space E
of infinitely many dimensions where linear functionals need not be continuous.
Those functionals form an unwieldy algebraic dual
which can be a monster. Instead we consider only the continuous dual
or topological dualE*, called dual for short,
which consists only of the continuous linear functionals.
(Many authors insist on using the notation E* for the algebraic dual
and E' for the topological dual. We don't.
No notation is needed for the algebraic dual, as it's never studied.)
A form (or covector) is an element
of the topological dual E*.
It's thus a continous linear functional when E is a space of functions.
Modern functional analysis
is essentially the study of the topological dualE*
of the vector space E, which may or may not be constructed itself as a space of functions.
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; ENS 1907)
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which might otherwise have been lost (Lévy had been entrusted in 1919 with the unpublished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 (1736-1813)
had inaugurated in 1794. Lévy had only
five 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...
By convention, the locution
Euclidean vector spaces
(sometimes opposed to affine Euclidean spaces)
only applies to spaces isomorphic to
n,
for some finite integer n. Euclidean spaces are thus finite-dimensional.
Over a complete scalar field K,
a finite-dimensional vector space is always complete. That need not be so for spaces with infinitely many dimensions.
Therein lies the basic challenge of functional analysis...
I remember attending a forceful lecture by Laurent Schwartz
in 1977 where he insisted that not much can be done in functional analysis
with a space which isn't complete,
because completeness is the key property which allows you
to assert the existence of a limit without the need for a candidate value a priori.
The methods of calculus
are normally applied only to complete versions
of the above, for which the following names are used:
(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]
®
u
®
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)
a
In fact, this equivalence is one way
Stieltjes measures could be introduced.
Those measures are arguably the forerunners of the general
distributionsLaurent Schwartz (1915-2002)
would devise in 1944
(actually, Stieltjes measures are just a special type of distributions).
Helly gave essentially the same proof as what Hans Hahn would publish for the general case 15 years later.
(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.
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...
(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 choice principle
(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
,
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
,
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)
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 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 E.
General proofs using weaker forms of Choice :
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:
Hahn-Banach Theorem for weaker vectorial structures :
The full structure of a Banach space isn't absolutely necessary to prove
the Hahn-Banach theorem. It can also be established for a
Fréchet space or even
just a locally convex topological vector spaces (LCTVS).
The same is true for the Kreine-Milman theorem.
(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
Lp [a,b] (p > 1)
of the complex functions of a bounded real variable.
However, his methods are readily applicable to any other complex linear space.
Bohnenblust &
Acknowledging that, Sobcyzk
(1938)
gave the Hahn-Banach theorem its final name and they popularized its complex version.
They also made it clear 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)
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.
(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.
(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.
(2021-08-19) Krein-Milman theorem (1940)
A compact convex set is the closed convex hull of its extreme points.
An extreme points M of a convex set K
is a point which is not halfway between two different points from that set.
" x Î K\{M} ,
" y Î K\{M} ,
M ¹ ½ x + ½ y
Relation with choice axioms :
A corollary, often called the Krein-Milman theorem is
that a nonempty compact convex set has at least one extreme point. That statement is
actually a weaker form of the
axiom of choice which is equivalent to the full axiom of choice
only in combination with some other particular choice axiom
(either the Hahn-Banach theorem or the ultrafilter lemma will do).
Restricted to finitely many dimensions, either version of the Krein-Milman theorem
can be proved in standard ZF set theory without using
any choice axiom. That was was done by
Minkowski in 1911 (in the 3D case)
or Ernst Steinitz in 1916
(for finite dimensionality).
Krein-Milman Theorem :
In a Hausdorff locally convex topological vector space X,
a compact convex subset K is equal to the
closure of the
convex hull of its extreme points.
Millman's "converse" :
If K is the closed convex hull of the compact T,
then every extreme point of K is in the closure of T.
p
® 
