Dieudonné 1950:
Grothendieck a tué l'analyse fonctionnelle (1:32:46)
by Pierre Cartier (in French, 2018年10月14日).
Grothendieck & Banach Spaces (1:58:22)
Gilles Pisier (French, 2018年02月06日).
Variétés stochastiques (1:43:31)
Anatole Khelif & Alain Tarica (2015年10月15日)
Functional Analysis Overview (49:35)
by Peyam Tabrizian (2018年10月04日).
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 space E 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 dual E*, 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 dual E*
of the vector space E, which may or may not be constructed itself as a space of functions.
Life, death and legacy of René Gateaux by Laurent Mazliak (arXiv, 2007年01月17日).
These are vector spaces endowed with a topology which makes both vector addition and scalar multiplication continuous functions for the relevant product topologies (assuming a standard topology for scalars).
This includes:
By convention, the locution Euclidean vector spaces (sometimes opposed to affine Euclidean spaces) only applies to spaces isomorphic to Rn, 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:
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Topological vector space
|
Bounded set (1935)
|
Uniform space (1937)
Lebesque spaces Lp
(Frigyes Riesz, 1910)
|
Banach spaces (1920)
Sobolev spaces
|
Sergei Sobolev (1908-1989)
Discontinuous linear map
|
Closed-graph theorem
|
Webbed space
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].
Every such functional can be uniquely expressed as a Stieltjes integral :
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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Stieltjes integration | Eduard Helly (1884-1943)
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Hahn-Banach Theorem
|
Hans Hahn (1879-1934)
|
Stefan Banach (1892-1945)
Riesz extension theorem (1923)
|
Marcel Riesz (1886-1969)
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:
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 signIn 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...
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:
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)
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. QED (Halmos symbol)
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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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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.
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年01月26日)
The Hahn-Banach theorem, Menger's theorem and Helly's theorem
by Terence Tao (2007年11月30日)
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).
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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
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年12月27日, 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
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)
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)
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Open mapping theorem (Banach-Schauder theorem) | Julius Schauder (1899-1943)
The closed unit ball of the dual space of a normed vector space is compact for the weak* topology.
The Bourbaki-Alaoglu theorem generalizes that to all dual topologies.
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Banach-Alaoglu theorem | Leonidas Alaoglu (1914-1981)
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.
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If K is the closed convex hull of the compact T, then every extreme point of K is in the closure of T.
>Krein-Milman theorem (1940) | Mark Krein (1907-1989) | David Milman (1912-1982)
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Stone-Weierstrass theorem (1937) | Marshall H. Stone (1903-1989; PhD 1926)
Most naturally occurring topological vector spaces are either Banach spaces or nuclear (they can't be both unless their dimension is finite).
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Schwartz kernel theorem (Schwartz, 1952) | Nuclear Spaces (Grothendieck, 1955)
A subset is called relatively compact when its closure is compact.
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Compact operator | Nuclear Spaces (Grothendieck, 1955)
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Schauder basis (1927) | Juliusz Schauder (1899-1943)
