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