Banach J. Math. Anal. 1 (2007), no. 1, 1–10

Banach Journal of Mathematical Analysis

ISSN: 1735-8787 (electronic)

http://www.math-analysis.org

ON STEFAN BANACH AND SOME OF HIS RESULTS

KRZYSZTOF CIESIELSKI1

Dedicated to Professor Themistocles M. Rassias

Submitted by P. Enﬂo

Abstract. In the paper a short biography of Stefan Banach, a few stories

about Banach and the Scottish Caf´e and some results that nowadays are named

by Banach’s surname are presented.

1. Introduction

Banach Journal of Mathematical Analysis is named after one of the most out-

standing mathematicians in the XXth century, Stefan Banach. Thus it is natural

to recall in the ﬁrst issue of the journal some information about Banach. A

very short biography, some of his most eminent results and some stories will be

presented in this article.

2. Banach and the Scottish Caf´

e

Stefan Banach was born in the Polish city Krak´ow on 30 March, 1892. Some

sources give the date 20 March, however it was checked (in particular, in the

parish sources by the author ([12]) and by Roman Ka lu˙za ([26])) that the date

30 March is correct.

Banach’s parents were not married. It is not much known about his mother,

Katarzyna Banach, after whom he had a surname. It was just recently discovered

(see [25]) that she was a maid or servant and Banach’s father, Stefan Greczek, who

Date: Received: 14 June 2007; Accepted: 25 September 2007.

2000 Mathematics Subject Classiﬁcation. Primary 01A60; Secondary 46-03, 46B25.

Key words and phrases. Banach, functional analysis, Scottish Caf´e, Lvov School of mathe-

matics, Banach space.

1

2 K. CIESIELSKI

was a soldier (probably assigned orderly to the oﬃcer under whom Katarzyna was

a servant) could not marry Katarzyna because of some military rules. Banach

was grew up with the owner of the laundry Franciszka P lowa and her niece Maria.

Banach attended school in Krak´ow and took there ﬁnal exams. He was in-

terested in mathematics and he was very good at it. However, he thought that

in this area nothing much new can be discovered and he decided to study at

the Technical University at Lvov (Politechnika Lwowska). In those times both

Krak´ow and Lvov were in the territory governed by Austro-Hungary. Therefore

Banach moved to Lvov. His studies there were interrupted by the First World

War and Banach came back to Krak´ow.

As a mathematician, Banach was self-taught. He did not study mathematics,

however he attended some lectures at the Jagiellonian University, especially de-

livered by Stanis law Zaremba. In 1916, a very important event took place. Hugo

Steinhaus, an outstanding mathematician, then already well known, spent some

time in Krak´ow. Once, during his evening walk at the Planty Park in the centre

of Krak´ow, Steinhaus heard the words “Lebesgue integral”. In those times it

was a very modern mathematical term, so Steinhaus, a little surprised, started to

talk with two young men who were speaking about the Lebesgue measure. These

two men were Banach and Otto Nikodym. Steinhaus told them about a problem

he was currently working on, and a few days later Banach visited Steinhaus and

presented him a correct solution.

Steinhaus realised that Banach had an incredible mathematical talent. Stein-

haus was just moving to Lvov where he got a Chair. He oﬀered Banach a position

at the Technical University. Thus Banach started his academic career and teach-

ing students, however he did not graduate.

Steinhaus used to say that the discovery of Banach was his greatest mathe-

matical discovery. It must be noted here that many outstanding mathematical

results are due to Steinhaus.

There is a curious story how Banach got his Ph.D. He was being forced to

write a Ph.D. paper and take the examinations, as he very quickly obtained

many important results, but he kept saying that he was not ready and perhaps

he would invent something more interesting. At last the university authorities

became nervous. Somebody wrote down Banach’s remarks on some problems,

and this was accepted as an excellent Ph.D. dissertation. But an exam was

also required. One day Banach was accosted in the corridor and asked to go

to a Dean’s room, as “some people have come and they want to know some

mathematical details, and you will certainly be able to answer their questions”.

Banach willingly answered the questions, not realising that he was just being

examined by a special commission that had come to Lvov for this purpose.

In 1922 Banach, aged 30, was appointed Professor at the Jan Kazimierz Uni-

versity at Lvov. After the First World War Poland got back its independence

and Krak´ow and Lvov were again in Poland.

In fact, Banach was interested in nothing but mathematics. He wrote down

only a small part of his results. He was speaking about mathematics, introducing

new ideas, solving problems all the time. Andrzej Turowicz, who knew Banach

very well, used to say that two mathematicians should have followed Banach all

ON STEFAN BANACH AND SOME OF HIS RESULTS 3

the time and written down everything he said. Then, probably the majority of

what Banach did would have been saved. Nevertheless, his results are incredible.

Some of them will be recalled in the sequel of this paper. For more details, see

[11], [13].

An important role in mathematicians’ life in Lvov was played by the Scottish

Caf´e (see [10]). It was a place of their meetings, where they were eating, drinking,

speaking about mathematics, stating problems and solving them. They used

to write solutions on marble tables in the caf´e. However, after each such visit

the tables were carefully cleaned by the staﬀ. Probably some diﬃcult proofs of

important theorems disappeared in this way. Therefore after some time Banach’s

wife, Lucja, bought a special book (called later the Scottish Book) that was

always kept by the waiters and given to a mathematician when ordered. The

problems, solutions, and rewards were written down in the book. One reward

became particularly famous. On 6 November 1936 Stanis law Mazur stated the

following problem (in the Scottish Book the problem had the number 153).

Problem 2.1. Assume that a continuous function on the square [0,1]2and the

number ε > 0 are given. Do there exist numbers a1, . . . , an,b1, . . . , bn,c1, . . . , cn,

such that

f(x, y)−

n

X

k=1

ckf(ak, y)f(x, bk)

≤ε

for any x, y ∈[0,1]?

Now it is often said that the problem was about the existence of Schauder basis

in arbitrary separable Banach space. However, that was not known at that time.

It was only in 1955 that Alexandre Grothendieck showed ([19]) that the existence

of such numbers is equivalent to so called “the approximation problem”, i.e. the

problem if every compact linear operator Tfrom a Banach space Xinto a Banach

space Yis a limit in norm of operators of ﬁnite rank. The problem was especially

attractive as Mazur oﬀered a prize: a live goose. The approximation problem

(and, consequently, the original Mazur’s problem) was solved only in 1972 by

a Swedish mathematician Per Enﬂo (then 28 years old) ([17]) who shortly after

giving a solution came to Warsaw and got from Mazur the prize.

The former Scottish Caf´e in 2006 (now a bank)

4 K. CIESIELSKI

There are plenty of stories about mathematicians in the Scottish Caf´e. Here,

let me recall two of them.

Once Mazur stated a problem and Herman Auerbach started thinking it over.

After a while Mazur said that, to make a puzzle more interesting, he oﬀered a

bottle of wine as a reward. Then Auerbach said: “Ah, so I give up. Wine does

not agree with me”.

The second story is connected with Lebesgue’s visit to Lvov in 1938. Lebesgue

came to Lvov, delivered a lecture and after that was invited to the Scottish Caf´e.

A waiter gave Lebesgue the menu. Lebesgue, who didn’t know Polish, studied

the menu for a while, gave it back and said: “I eat only dishes which are well

deﬁned”.

The period since the end of the First World War up to 1939 was the Golden

age for Polish mathematics. In particular, in those times time Banach obtained a

series of remarkable results. Banach can be regarded as the creator of functional

analysis, which at that period could be named “the Polish branch of mathemat-

ics”. In Lvov, together with Banach, there worked many outstanding mathemati-

cians, for example Steinhaus, Mazur, Juliusz Schauder, W ladys law Orlicz, Stan

Ulam and Mark Kac (the last two moved to the USA before the war). Polish

mathematicians worked actively also in other centers, especially in Warsaw; for

example, Wac law Sierpi´nski, Karol Borsuk, Kazimierz Kuratowski and Stefan

Mazurkiewicz should be mentioned.

Banach and Steinhaus initiated in 1928 a new journal, “Studia Mathematica”

which published papers just on functional analysis. It was one of the ﬁrst journals

in history that was specialized in some particular areas of mathematics (the very

ﬁrst one was “Fundamenta Mathematicae”).

In 1931 the fundamental monograph on functional analysis by Banach was

published. It was “Operacje liniowe”, in 1932 published in French “Th´eorie

des op´erations lin´eaires” as the ﬁrst volume in the series “Mathematical Mono-

graphs”. For many years it was the most basic book on functional analysis, up

to the moment when the famous monograph [16] was published (see also [30]).

It should be noted that only in 40s the term “functional analysis” was being

used. In Banach times other names were of use, especially “the theory of linear

operators”.

In 1939 Lvov was captured by the Soviet Union, in 1941 Hitler’s soldiers took

Lvov for 4 years. Banach spent the Second World War in Lvov, living under

extremely diﬃcult conditions. After the war Lvov was taken by the Soviet Union

again and Banach planned to go to Krak´ow where he would have taken a Chair

at the Jagiellonian University. He died just a few days before the move. He is

buried in Lychakov Cemetery (Cmentarz Lyczakowski) in Lvov. Now, in front of

the building of the Mathematics Institute of the Jagiellonian University there is

a monument of Banach.

ON STEFAN BANACH AND SOME OF HIS RESULTS 5

The monument of Banach in Krak´ow

The Scottish Book survived the war. It was taken to Poland by Lucja Banach

and later on translated to English by Steinhaus. Ulam let the problems from the

book circulate in the United States. In 1981 the book was published in English

in the version prepared by Dan Mauldin ([27]). This translation is remarkably

valuable as besides the problems and solutions (if there are) it includes several

comments and remarks about the continuation of the investigations inspired by

the problems from the Scottish Book. It is a large and important piece of math-

ematics.

The international mathematical centre in Poland, created in 1972 is named

after Banach. The Banach Center is a part of the Institute of Mathematics of the

Polish Academy of Sciences and has its main oﬃce in Warsaw. Conferences took

place in Warsaw, recently they have been organized also in B¸edlewo. There are

Banach Center Publications that publishes proceedings of selected conferences

and semesters held at the International Stefan Banach Mathematical Center. Up

to now, 77 volumes were published.

3. Some results named by Banach’s surname

Now we turn to some mathematical results of Banach. As mentioned above,

only some of his discoveries were published; nevertheless, they present themselves

an enormous collection. Here, we present only some of the most important results

which are nowadays named after him; the results will be recalled and it will be

said where they were published.

Before that, an important fact should be mentioned. It was checked what

names appear most frequently in the titles of mathematical and physical papers

published in the 20th century. It turned out that it was Banach’s name that

got the ﬁrst place. The second place was obtained by Sophus Lie, the third by

Bernhard Riemann.

Certainly, Banach deserves such a position mainly because of Banach spaces,

nowadays one of the most important mathematical notions. A Banach space is

a normed complete vector space. It was formally deﬁned in the paper [5]; for

some time mathematicians in Lvov called those spaces “a space of type B”. The

6 K. CIESIELSKI

name “Banach space” was probably used for the ﬁrst time by Maurice Fr´echet, in

1928. Note that independently such spaces were introduced by Norbert Wiener,

however Wiener thought that the spaces would not be of importance and gave

up. A long time later Wiener wrote in his memoirs that the spaces quite justly

should be named after Banach alone, as sometimes they were called “Banach-

Wiener spaces”. For more details, see [14] and [15].

There are some points which show why the introduction of Banach spaces

was so important. For a variety of reasons function spaces are very useful in

many investigations and applications. To a large extent, modern mathematics is

concerned with the study of general structures. The essential thing is ﬁnding the

right generalization. Insuﬃcient generality can be too restrictive and a great deal

of generality may result in a situation where little can be proved and applied.

The space introduced by Banach, especially pointing out completeness, attests to

his genius; he hit the traditional nail on the head.

Banach’s great merit was that, in principle, it was thanks to him that the

“geometric” way of looking at spaces was initiated. The elements of some general

spaces might be functions or number sequences, but when ﬁtted into the structure

of a Banach space they were regarded as “points”, as the elements of a “space”.

At times this resulted in remarkable simpliﬁcations.

Today, almost ninety years after its introduction, the notion of a Banach space

remains fundamental in many areas of mathematics. The theory of Banach spaces

is being developed to this day, and new, interesting, and occasionally surprising

results are obtained be many researches. In particular, some really important

results were obtained in the end of the 20th century by William Timothy Gowers.

Some problems he solved waited for the solution since Banach’s times. For his

research, Gowers was awarded in 1998 with the Fields medal.

In the same paper [5] there is proved the famous Banach Fixed Point Theorem.

It says

Theorem 3.1. Let (X, d)be a complete metric space and a function f:X→X

be a contracting operation, i.e. there exists a λ∈(0,1) such that d(f(x), f(y)) ≤

λd(x, y)for any x, y ∈X. Then there exists a unique p∈Xsuch that f(p) = p.

The theorem was the use of the method of successive approximations and a

general version of the property that was known earlier in some concrete cases.

Now we turn to some fundamental theorems on functional analysis.

One of the most important of them is the Hahn-Banach Theorem.

Theorem 3.2. Let Xbe a real normed vector space, p:X→Ra function such

that p(αx + (1 −α)y)≤αp(x) + (1 −α)p(y)for each x, y ∈Xand α∈[0,1].

Assume that ϕ:Y→Ris a linear functional deﬁned on a vector subspace Y

of Xwith the property ϕ(x)≤p(x)for all x∈Y. Then there exists a linear

functional ψ:X→Rsuch that ψ(x) = ϕ(x)for all x∈Yand ψ(x)≤p(x)for

all x∈x.

The theorem has several versions (compare [29]).

It was published by Banach in [3]. Independently, it was (in a simper case)

slightly earlier discovered by Hans Hahn ([20]) which was a generalization of

ON STEFAN BANACH AND SOME OF HIS RESULTS 7

his result from 1922. Banach did not know about Hahn’s paper; neverthelesss,

Banach’s version was stronger as Hahn proved a theorem in the case where X

is a Banach space. It should be noted that a simpler version of the theorem (in

the case where Xis the space of continuous real functions on a compact interval)

was published much earlier by Eduard Helly ([21]; see [22]). Neither Banach nor

Hahn knew about Helly’s theorem.

The complex version of the theorem was proved later, in 1938 independently

by G.A. Soukhomlinov and by H.F.Bohnenblust ([8]) and A.Sobczyk ([31]).

The Hahn-Banach Theorem is regarded by many authorities as one of three

basic principles of functional analysis. The two others are the Banach-Steinhaus

Theorem and the Banach Closed Graph Theorem.

The Banach-Steinhaus Theorem was proved in [7]. It says:

Theorem 3.3. Let Xbe a Banach space, Ybe a normed vector space. Consider

the family Fof all linear bounded functions from Xto Y. If for any x∈Xthe

set {||T(x)|| :T∈ F} is bounded, then the set {||T|| :T∈ F} is bounded.

Now recall the Banach Closed Graph Theorem.

Theorem 3.4. Let Xand Ybe Banach spaces, and Tbe a linear operator from

Xto Y. Then Tis bounded if and only if the graph of Tis closed in X×Y.

This theorem is closely related to the very important Banach Open Mapping

Principle.

Theorem 3.5. Let Xand Ybe Banach spaces, and Tbe a linear operator from

Xonto Y(we assume that Tis surjective). Then for any open subset Uof X the

set T(U)is open in Y.

Both theorems were published in [6].

Let us mention also another theorem on functional analysis, nowadays called

frequently the Banach-Alaoglu Theorem. It says

Theorem 3.6. Let X∗be the dual space of a Banach space X. Then the closed

unit ball in X∗is compact in X∗with the weak–∗topology.

A proof of this theorem was given in 1940 by Leonidas Alaoglu ([1]) and in the

case of separable normed vector spaces was published in 1929 by Banach ([4]).

Banach did not work only on functional analysis. For example, today his name

is connected with famous Banach-Tarski Theorem on paradoxical decomposition

of the ball. The theorem may be formulated in the following way.

Theorem 3.7. If B⊂R3is a three-dimensional ball, then there exist pairwise

disjoint sets A1, . . . , Anand isometric transformations I1, . . . , Insuch that B=

A1∪. . . ∪An, and for some k∈(1, n):I1(A1), . . . , Ik(Ak)are pairwise disjoint,

B=I1(A1)∪. . . ∪Ik(Ak),Ik+1(Ak+1), . . . , In(An)are pairwise disjoint, B=

Ik+1(Ak+1 )∪. . . ∪In(An)

The theorem looks very strange, as it, in fact, says that we can double the

volume! The point is that the ball is split into pieces that are non-measurable.

The proof relays on the axiom of choice. It was published in [9]. For more

information, see [18].

8 K. CIESIELSKI

Not all the mathematical theorems and notions which are now frequently called

by Banach’s name were decribed above. Let us mention here, for instance, Banach

integral, Banach generalized limit (introduced in [2]) and Banach algebra. Banach

algebras were a kind of restructurization of Banach spaces (instead of a vector

space there is taken a ring and in addition a multiplication of elements). Banach

algebras were introduced in 1941 by a Russian mathematician Israil M. Gelfand.

One should mention here also the Banach-Mazur distance (introduced in [6]),

which is a suitable deﬁned distance between two isomorphic Banach spaces.

The ﬁrst volume of the Banach Journal of Mathematical Analysis is dedicated

to Themistocles M. Rassias. It is nice to notice that some of the achievements of

Th.M. Rassias have a particular connection with Banach and the mathematics

from the Scottish Caf´e.

One of the most important mathematicians of the Lvov group was Stan Ulam,

who was very young in his Lvov days. In 1936 he moved to the USA where he

later on became a very famous scientist. The reader is referred to the wonderful

volume [32]. As was mentioned above, Ulam played a great role in circulating

the mathematics from the Scottish Caf´e after the Second World War.

With the names of Ulam and Rassias there is connected a mathematical term,

now widely known as Ulam–Hyers–Rassias stability. Let Xand Ybe real Banach

spaces. The stability of Ulam–Hyers–Rassias approximate isometries on restricted

domains S(bounded or unbounded) for into mapping f:S→Ysatisfying

||f(x)−f(y)|| − ||x−y|| ≤ εespecially where Yis a Banach space.

In 1940 Stan Ulam stated the problem concerning the stability of homomor-

phisms: Let G1be a group and let G2be a metric group with a metric dand let

ε > 0be given. Does there exist a δ > 0such that if a function h:G1→G2

satisﬁes the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈G1, then there exists

a homomorphism H:G1→G2with d(h(x), H(x)) < ε for all x, ∈G1?Roughly

speaking: When does a linear mapping near an “approximately linear” mapping

exists? T.M. Rassias gave a solution in [28], introducing some condition for map-

pings between Banach spaces. A particular case of Rassias’s theorem was the

result of Donald H. Hyers ([23]). Now it can be said that the study of Ulam–

Hyers–Rassias stability in its present form was started by the paper of Th.M.

Rassias [28]. For more information of such kind of stability and the basic papers

on the subject, see [24].

Let us end with some more anecdotes.

There is a story to the eﬀect that, upon publication, Banach’s monograph

“Theory of operations. Linear operations” was displayed in some Lvov bookshops

on shelves labelled “Medical Books”.

In some Polish “cities, including Krak´ow and Warsaw, there are streets named

“Banach street”. In Warsaw, now the Mathematical Institute of Warsaw Univer-

sity has its house at Banach street. The 1983 International Congress of Mathe-

maticians took place in Warsaw. A few foreign mathematicians found out that

there is a street in Warsaw called Banach Street, and this is the last stop on a

certain trolley line. Curious about Banach Street, they got on the trolley, got oﬀ

ON STEFAN BANACH AND SOME OF HIS RESULTS 9

at the last stop, and were confronted by a sizable empty area. They arrived at

the unanimous conclusion that what they were facing was not “Banach street”

but rather a “Banach space”.

The list of references is far from complete, as there are enormous numbers of

papers connected with Banach. Here, except the original papers mentioned in the

article, are given only some papers in English where the reader can ﬁnd several

additional information.

Acknowledgements: The photographs were taken by Danuta Ciesielska and

Krzysztof Ciesielski.

References

1. L. Alaoglu, Weak topologies of normed linear spaces, Ann. Math. 41(1940), 252–267.

2. S. Banach, Sur le probl´eme de la mesure, Fund. Math. 4(1924), 7–33.

3. S. Banach, Sur les fonctionelles lin´eaires, Studia Math. 1(1929), 211–216.

4. S. Banach, Sur les fonctionelles lin´eaires II, Studia Math. 1(1929), 223–239.

5. S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations

int´egrales, Fund. Math. 3(1922), 133–181.

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10 K. CIESIELSKI

24. D.H. Hyers and T.M. Rassias, Approximate homomorphisms, Aequationes Math., 44(1992),

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Soc. 72(1978), 297–300.

29. W. Rudin, Functional analysis, second edition, McGraw-Hill, 1991.

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1Mathematics Insitute, Jagiellonian University, Reymonta 4, 30-059 Krak´

ow,

Poland.

E-mail address:Krzysztof.Ciesielski@im.uj.edu.pl