DAVID
MILMAN MEMORIAL
ISSUE
DAVID MILMAN
(1912-1982)
0378-620X/86/010001-1051.50+0.20/0 1986 Birkh~user Verlag, Basel
Integral Equations and Operator Theory Vol. 9 (1986)
DAVID M I L M A N
On 12th July, passed
away.
analysis
Milman
Passover,
Ukraine,
For the
recorded
began
education
his
secular
Hebrew
both o r i g i n a l
included
the works
a Ukrainian
(called
the
Unfortunately School
owned
In 1929
about
Ivan T i m c h e n k o , University,
who helped
theorem,
university. second
Fermat's
David
but
works
D. M i l m a n
both r e l i g i o u s reading
(amazingly,
At the
father
were
There
interested
he also
viewed
store
before
in mathematics. Professional
unfavourably
the R u s s i a n
revolution.
Once there,
Last Theorem.
Soon a f t e r w a r d s at Odessa
him find an error
student
that
David
weekly magazine,
of M a t h e m a t i c s
suggested
by the
of C h i c h e l n i k
to Odessa.
in a p o p u l a r
college
he met
Chemistry
was the rabbi
in
these
same time
School).
the
near Vinitza,
i, 1913.
he r e c e i v e d
before
the local
age 14, c o n t i n u e d
to leave the
was a c c e p t e d
year m a t h e m a t i c s
Saturday
Chichelnik
they became
an a r t i c l e
Professor
contributions.
and from age 14 a c h e m i s t r y
origins
a shoe
David M i l m a n
in f u n c t i o n a l
"simplicity,"
where until
his family moved
came across
"Ogonyok,"
Fermat's
social
the
as J a n u a r y
Professional
his m o t h e r ' s
and his f a t h e r
by chance
school,
he was forced his
sake of
of Shakespeare).
and t o g e t h e r
because
new regime:
in 1912,
and t r a n s l a t e d
Chemistry
Gelfand
and,
pioneers
town called
in Heder,
instruction
Professor
for his o u t s t a n d i n g
his b i r t h d a y
attended
Israel
in Tel Aviv,
was born
in a small
now USSR.
registrar
and
- 1982)
He was one of the famous
and he is r e m e m b e r e d David
Jewish
1982,
(1912
in his
he should
he met
State
"proof"
of
apply to the
one m o n t h
later
(1931)
as a
at Odessa
State University.
He
2
graduated
in 1934.
authorities
Again,
.
of his
Krein.
1937, when he became
He received
years of functional first
because
V
LIvslc
did not allow him to proceed
edueation until M.G.
9
Gohberg,
seminars
his Ph.D.
analysis.
in functional
and Piatetski-Shapiro
social origins,
a graduate
in 1939.
M.G.
student
in the USSR
David Milman was one of the active participants Over various included
periods
V.R. Gantmacher, V.P.
of time the participants
V.A. Artemenko,
Potapov,
B.Ja.
M.A.
M.S.
Brodskii,
Levin, M.S.
Ruttman,
V.L.
produced
and discussed
abstract
operator
theory
applied
problems
tions.
David Milman
in the
Livsic,
M.A.
how the Brodskii/Milman was proved.
seminar. seminar
Naimark,
analysis
seminar.
in Banach
and engineering by members
were
The
spaces
and up to
and their connecof the seminar
source of new ideas.
as
It is recalled
theorem on the center of a convex
living
in the Dachas
in the resort
set
which are on the outskirts
area on the Black Sea coast.
morning David Milman would visit new idea of the proof.
Brodskii
They would
Every
and bring with him a
sit together
and discuss
the proof until they found that the idea they were working would not work. eonsecutive
and
Milman and Brodskii worked during the summer
vacation whilst of Odessa
of this
with geometry
is remembered
and
Many contributions
(linear and non-linear)~
in mechanics
a man who was a constant
(Odessa),
F.R. Gantmacher,
Smul'jan.
by the members
stirting
one of the
in this
which now form a part of the basis of functional
topics were varied,
of
These were the boom
Krein was leading
analysis
the
with his mathematical
This procedure
days,
was repeated
until on the thirtieth
on
for twenty-nine
day, the thirtieth
idea went through. Unfomtunately,
soon after World War II this
and the school of functional
analysis
at Odessa University,
to an end because M.G.
Krein was dismissed
to leave.
was very
The seminar
seminar
important
and his
students
for David Milman
came had as
a forum in which he could try out his ideas. David Milman was very devoted to mathematics, research and education, in either
space or time.
and this dedication
knew no bounds,
On one occasion he came to M.G.
Krein
9
Gohberg,
V,
LIvslc
and P i a t e t s k i - S h a p i r o
after m i d n i g h t mathematical
idea.
be a f a c t o r protest
he had to catch
Together
they raised
he c o n t i n u e s
Russia, worked
works
on g e o m e t r y
He joined
in r e v i s i n g
and he served 1949-1966. offered
there
beginning
with criteria
every
s > 0
x + yiI
This t h e o r e m
He also
the unit
spaces,
Acad.
the
collection
sphere
eight
Sci. U.S.A.
David
= IIyll
following
study
ball
that
convex; such that IIx-ylI
< 6 .
of w h i c h
and
continues
results
to follow.
sufficient
is w e a k l y
closed
to
convex
compact
subsets
and
of
intersection. [8],
MR i,
(1947),
He proved
> 0
for other
the unit
deal
in the G e o m e t r i c
necessary
of n o n e m p t y
Milman
later.
(1938-1939)
implies
result
immediately
f r o m the very
spaces.
= i,
the first
from
He a r r i v e d
years
6 = 6(6)
the a c t i v e
(Russian) 33
and was
if it is u n i f o r m l y
exists
has a n o n e m p t y
317-328
institution
Department
of D. M i l m a n
but also an i n s p i r a t i o n
Later, (1939),
IIxiI
for r e f l e x i v i t y ;
nested
an important
in H e b r e w
of Banach
there
not o n l y
established
condition every
was
of Banach
the present,
passing
is r e f l e x i v e
> i - 6,
Polytechnic
at this
to Israel
and lectured
in
of C o m m u n i c a t i o n s
He played
program
is a is still
to 1945 David
at the
at Tel A v i v University.
for r e f l e x i v i t y
i.e.
theory
1974.
who
at Tel
analysis,
Peter
F r o m lg3g
Institute
publications
E
spaces.
of the M a t h e m a t i c s
his u n t i m e l y
space
89
until
fluently
a Banach
all of w h o m are
(Dozent)
the m a t h e m a t i c s
The first
for
science.
In 1974 he immigrated
until
a physics
of m a t h e m a t i c s
and Vladimir,
the Odessa
as c h a i r m a n
Hebrew
the next
Tsudikova,
sons,
of Banach
Professor
a professorship
speaking
should
Krein's
to w o r k on f u n c t i o n a l
University,
in c o m p u t e r
in 1945 and w o r k e d role
Nemo
is a p r o f e s s o r
as an A s s o c i a t e
Institute.
Even
an early t r a i n
three
A v i v University;
at T o r o n t o
important
why the hour
discussion.
David m a r r i e d
Vitali
professor
very
on Milman.
mathematicians.
and e s p e c i a l l y
and for him,
not u n d e r s t a n d
immediate
had no effect In 1938,
major.
a fresh,
He could
preventing
that
morning,
to d i s c u s s
3
Vitold
335],
51-53.
MR
W.F.
Smul'jan
[Mat.Sb.5(47)
Eberlein
g, 42; MR 10,
[Proc. 855],
Nat. and
,
4
V
.
Gohberg, LlVslC and Piatetski-Shapiro
also David Milman together with Vitali Milman [23, 25, 26] found even stronger criteria for reflexivity of Banach spaces. In his work with B.Ja. Levin [4], David Milman proved that the
only closed subspaces of both the Banach space of
functions of bounded variation and the space of continuous functions on [0, i] are finite dimensional
subspaces.
The same year (1940) he published with M.G. Krein the well-known Krein-Milman theorem [3]. points of a convex set
K
K
extreme points of
extreme
are the points that do not belong to
interior of any interval in every point of
By definition,
K
In a finite dimensional space
is a centre of mass distributed among the K
The famous Krein-Milman theorem says:
Every bounded regularly convex subset
K
of a dual to
a Banach space is the regularly convex hull of its extreme points. This theorem entered all text books in functional analysis in the following form due to Bourbaki: set in a separable
a compact convex
locally convex space is the closed convex ~ull
of its extreme points.
It was a starting point of the "Method of
extremal points and centres" developed by D. Milman in the 1940's and early 50's in [8,10,13,17,193, doctoral thesis
in his Russian second
(1951) and later in a monograph [31].
This
theory has been further developed by G. Choquet, V. Klee and others
(see [31]).
The Krein-Milman theorem and "the method of
extreme points and centres" have important applications.
Let us
mention a few of them: The theorem of I.M. Gelfand and D.A. Raikov on the existence of complete systems of irreducible unitary representations of locally compact groups [Mat. Sb. 13 (55) 316~ Amer. Math.
Soc. Transl.
(1943),
301-
(2) 36 (19~4), 1-15. MR 6, i47]~
M.G. Krein's representation theorems for positivedefinite functions [Amer. Math.
Soc. Transl.
(2) 34 (1963),
69-164. MR 14, 480 and MR 12, 719]; M.A. Naimark's Nauk 3:5 (27),
(1948),
"Rings with involution" [Uspechi Math.
52-145] and I.M. Gelfand and M.A. Naimark's
"Normed rings with involution and their representations"
[Izv.
Akad. Nauk SSSR Ser. Mat. 12 (1948), 445-480 MR i0, 199]
(see
9
Gohberg,
V.
LIVSlC and P i a t e t s k i - S h a p i r o
5
also D, M i l m a n [8~ iI])~ D. M i l m a n ' s t h e o r e m on the existence measure
in d y n a m i c a l
of invariant
systems defined by f u n c t i o n a l s
[12].
Inspired by the theory of normed rings and the m e t h o d of extreme points, "T-boundary." fixed
set
R
all points of
q0
'
D. M i l m a n [8,9]
The T - b o u n d a r y of continuous
q06Q
R
functions
f(q)
>
or
If(q)l
closed
for
fs
subset
functions
functions
space
Q
with a
is the set
T
Q
such that f(q)
.
on
Q
one can consider
For example,
for a convex
of a dual to a Banach space with Q ;
Q
is the usual b o u n d a r y of several v a r i a b l e s
T-boundary
is the skeleton of
T-boundary
is the closure
the T - b o u n d a r y
for a family
on a domain
that for at least one
of U(q0)
o)
functions
of the extreme points of T-boundary
Q
Sup
being linear functionals,
analytic
on
f 6 R
qs
In case of c o m p l e x - v a l u e d
the
of a t o p o l o g i c a l
there exists a f u n c t i o n
regularly
the notion of
such that for every open n e i g h b o u r h o o d
Sup q6U(q 0 )
Re f(q)
introduced
R
is the closure of harmonic
in the complex plane, of
Q;
family
(in a d o m a i n Q .
R
f 6 R
the
In the separable
and for all
D. M i l m a n proved that every f u n c t i o n
of analytic
Q ccn),
of the set of all points
(or
the
case, the q06Q
such
q ~ q0' f(q0 ) > f(q)"
f 6 R
admits an integral
representation f(q0 ) = ff(q) d~(I ,q00) T q with m e a s u r e ~(Iq,q0) supported on uniquely
in the separable
subspaces
paper
of a Banach space was i n t r o d u c e d w i t h a small gap.
study of defect numbers angular
operators
by
q0
[14] w r i t t e n by D. M i l m a n
Krein and M. K r a s n o s e l s k i i ,
have been proved w h i c h e s t a b l i s h subspaces
and d e t e r m i n e d
case.
In the important jointly with M.
T
and studied.
e q u a l i t y of d i m e n s i o n s
These r e s u l t s
of linear operators
and others.
the gap b e t w e e n two Theorems for
are t h e n used for the in Banach
spaces,
This paper played an important
role later in the t h e o r y of F r e d h o l m operators.
6
Among
other results,
I. S h a f a r e v i c h ' s commutative logical zation
criterion
topological
rings
of d e n s e
subrings
K
set.
structure,
t h e n there
is a c o m m o n
surjeetive
isometries
of
for a larger
is a convex,
class
He i n t r o d u c e d Banach
space
x, y s K
E;
and
IIxll
=
llyll
of cones cone
in B a n a c h
exist
fl
and
existence tinuous
D. M i l m a n
theorems. additive
[24,30]
sublinear It I
subadditive p(t)
=
are
(1979-81),
special
of an open d o m a i n 35].
of
of
~n
,
has
cases
It t u r n e d
of the t h e o r y
it also
of the
f E E*
there
implies
the
identifies
of conK
w i t h the
for
sublinear
on
the
~i
the
dimension,
and
studied
subset
These results
theorems
B of [32].
the n o t i o n
set of all namely
symmetric containing
Also
in Israel
its s t r u c t u r e
G
all
of f u n c t i o n a l
of the c e n t r a l
Q = ~G of
Banach
to t h e case of sub-
set of all
Numerous
of T h e o r e m
the central
fundamental
is one d i m e n s i o n a l ,
infinite
~n
theorem
the t h r e e
while
introduced
Q, see [35].
and
in a if
> 6
for every
this r e s u l t
p
~ > 0 .)
in
K
normality
an e x t e n s i o n
In the case that t h e b o u n d a r y
manifold locus
p
G
cone
with a subspace
(Note that
k ~ 0 ,
D. M i l m a n
, E
this
mappings).
such that
Krein,
set, w h i c h
implies
[32J.
for
K
set of all
i(E).
functions
functions
for the
IIx+yl]
that
of
he extended
for
[sin ~.t I
analysis
in
which
functionals
= X
i
also found
In Israel
symmetric p(t)
from
on a compaet
functions
functionals
then
to the c o n d i t i o n
f2 = f - fl
functions
point
t h e y have proved
~ > 0
set
set w i t h a normal
of a normal
due to M.G.
D. M i l m a n
of a c o n v e x
nondecreasing
exists = i
of an i s o m o r p h i s m
nonnegative
fixed
of c o n t i n u -
Brodskii,
w-compact
topo-
a characteri-
rings
By the two m a i n t h e o r e m s
spaces
is e q u i v a l e n t
implies
structure
of d i s t a n c e
there
out to be v e r y useful:
this
(moreover,
the n o t i o n
i.e.,
of a n o r m on a
Jointly with
that
K
[7] e x t e n d e d
topological
of a normal
and proved
theorem
if
and P i a t e t s k i - S h a p i r o
to the case of c o m m u t a t i v e
of c o m p l e t e
the n o t i o n
,
existence
In p a r t i c u l a r
on a compact
introduced
V
LlVSlC
D. M i l m a n
for t h e
field
w i t h unit.
ous f u n c t i o n s [9]
,
Gohberg,
subset [33,34,
is a smooth is R. T h o m ' s
subcut
of D. M i l m a n have a l r e a d y
,
Gohberg,
v
.
Llvslc and P i a t e t s k i - S h a p i r o
found r e c o g n i t i o n introductions Euclidean
in s i n g u l a r i t y
to J.N. M a t h e r ' s
space,"
Proc.
and Y. Yomdin's
central
set," C o m p o s i t i o Math.
manuscript
theory
(see, for example,
"Distance
of Symposia
199-215
7
from a submanifold
in Pure Math.
"On the local
structure
43 (1981),
[36], w h i c h is to appear
the
vol.
40
in
(1983),
of a generic
225-238).
in this issue,
His last is a continu-
a t i o n of his earlier work in [20,27]. David M i l m a n had a p h i l o s o p h i c a l t e n d e n c y towards extraordinary tion.
simple,
lecturer,
general and fruitful
ideas.
and a He was an
d e v o t i n g m u c h time and effort to educa-
He also was a kind and sympathetic
help to t h o s e
inclination
man who always
offered
in need.
I. Gohberg
M.S.
Llvsmc
I. P i a t e t s k i - S h a p i r o