# Władysław Orlicz

### Quick Info

Okocim, Galicia, Austria-Hungary (now Poland)

Poznań, Poland

**Władysław Orlicz**was a Polish mathematician who worked in functional analysis and topology.

### Biography

**Władysław Roman Orlicz**was born in Okocim, a village near Cracow. His parents, Franciszek and Maria née Rossknecht, had five sons. His father died when he was only four years old.

In 1919 Orlicz's family moved to Lvov (Lwów in Polish which is now Lviv in Ukraine), where he completed his secondary education and then studied mathematics at the Jan Kazimierz University in Lvov having Stefan Banach, Hugo Steinhaus and Antoni Lomnicki as teachers.

From 1922 to 1929 he worked as a teaching assistent at the Department of Mathematics of Jan Kazimierz University in Lvov. In 1928 he wrote his doctoral thesis

*Some problems in the theory of orthogonal series*under the supervision of Eustachy Żyliński. In the same year he married Zofia Krzysik. In the late twenties and early thirties Orlicz worked as a teacher in private secondary schools and in a military school.

Orlicz spent the academic year 1929/30 at Göttingen University on a scholarship in theoretical physics, not in mathematics. During his stay in Göttingen he started his collaboration with Zygmunt Wilhelm Birnbaum (also from Lvov). They published two papers in

*Studia Mathematica*in 1930 and 1931. Their results became a starting point for Orlicz to consider and investigate in 1932 and 1936 function spaces more general than $L^{p}$ spaces which later on became known as

*Orlicz spaces*.

It should be emphasized that from the functional analysis point of view (that is, as function spaces) Orlicz spaces appeared for the first time in 1932 in Orlicz's paper:

*Über eine gewisse Klasse von Räumen vom Typus B*Ⓣ in

*Bull. Int. Acad. Polon. Sci*. A 1932,

**8/9**, 207-220 with an additional condition on the function ( the so called $\Delta _{2}$-condition for large $u$), and in full generality (that is, without the $\Delta _{2}$-condition) in 1936.

In the years 1931-1937 Orlicz worked at the Lvov Technical University. In 1934 he was granted the habilitation (

*venia legendi*) for a thesis entitled

*Investigations of orthogonal systems*.

Working in Lvov Orlicz participated in the famous meetings at the Scottish Café (Kawiarnia Szkocka) where Stefan Banach, Hugo Steinhaus, Stanisław Ulam, Stanisław Mazur, Marek Kac, Juliusz Schauder, Stefan Kaczmarz and many others talked about mathematical problems and looked for their solutions. The group gained international recognition and was later described as the

*Lvov School of Mathematics*.

A collection of 193 mathematical problems from meetings at the Scottish Café appeared later on as the

*Scottish Book*. Orlicz is the author or co-author of 14 problems there. (R D Mauldin edited

*The Scottish Book, Mathematics from the Scottish Café*(Birkhäuser 1981) which contains problems and also commentaries on them by specialists).

You can see a picture of the Scottish Café at THIS LINK.

In 1968 when presenting the mathematical output of Steinhaus (in an article published in

*Wiadom. Mat*. in 1969), Orlicz wrote:

In Lvov under the leadership of our dear masters Banach and Steinhaus we were practising intricacies of mathematics.In 1937 Orlicz became a professor at Poznań University (now Adam Mickiewicz University) and spent the Second World War in Lvov. He was professor at the State University of Iwan Franko from January 1940 to June 1941 and from August 1944 to February 1945 he also taught at the school of commerce and handicrafts and lectured at forestry courses. In March 1945 Orlicz went back to Poland and in May 1945 he returned to University of Poznań. In July 1948 Orlicz was promoted to an ordinary professorship.

Until his retirement in 1974 he worked both at the University of Poznań and the Mathematical Institute of the Polish Academy of Sciences, Poznań Branch. Orlicz continued his seminar

*Selected Problems of Functional Analysis*until 1989. The seminar ran every Wednesday from half past twelve to two o'clock in Mathematical Institute. He was interested in works of other mathematicians and in branches far removed from functional analysis.

Orlicz collaborated with several mathematicians. Collaboration with Mazur was especially fruitful. They wrote a dozen joint papers and their results are now considered classical theorems. When in 1960 Steinhaus was writing about Banach he emphasised this fact (

*Nauka Polska*

**8**(4) (1960), 157 or

*Wiadom. Mat*.

**4**(1961), 257 or [2], 242):

Mazur and Orlicz are direct pupils of Banach; they represent the theory of operations today in Poland and their names on the cover of "Studia Mathematica" indicate direct continuation of Banach's scientific program.Altogether Orlicz published 171 mathematical papers, about half of them in cooperation with several authors. He was the supervisor of 39 doctoral dissertations and over 500 master's theses.

Orlicz participated in congresses of mathematics in Oslo (1936), Edinburgh (1958), Stockholm (1962) and Warsaw (1983), and in many scientific conferences. He was invited to universities in Canada, China, Germany and Israel.

His book

*Linear Functional Analysis*, (Peking 1963, 138 pp - in Chinese), based on a series of lectures delivered in German on selected topics of functional analysis at the Institute of Mathematics of the Academia Sinica in Beijing in 1958, was translated into English and published in 1992 by World Scientific, Singapore. Orlicz is also a co-author of two school textbooks.

Orlicz was the editor of

*Commentationes Mathematicae*(1955 - 1990), and of

*Studia Mathematica*(1962 - 1990), and President of the Polish Mathematical Society (1977 - 1979).

In 1956 Orlicz was elected a corresponding-member of the Polish Academy of Sciences and in 1961 its full member. Three universities (York University in Canada, Poznań Technical University and Adam Mickiewicz University in Poznań) conferred upon him the title of doctor

*honoris causa*, in 1974, 1978 and 1983, respectively.

Orlicz was awarded many high state decorations, prizes as well as medals of scientific institutions and societies, including the Stefan Banach Prize of the Polish Mathematical Society (1948), the Golden Cross of Merit (1954), the Commander's Cross of Polonia Restituta Order (1958), Honorary Membership of the Polish Mathematical Society (1973), the Alfred Jurzykowski Foundation Award (1973), Copernicus Medal of the Polish Academy of Sciences (1973), Order of Distinguished Teacher (1977), Wacław Sierpiński Medal of the Warsaw University (1979), Medal of the Commission for National Education (1983) and the Individual State Prizes (second degree in 1952, first degree in 1966).

Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly

*Orlicz spaces*), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.

Orlicz spaces $L^{\phi} = L^{\phi} (\Omega , \Sigma , \mu)$ are Banach spaces consisting of all $x \in L^{0}(\Omega , \Sigma , \mu)$ such that $\int _{\Omega } \phi(\lambda|x(t)|)d\mu(t) < ∞$ for some $\lambda = \lambda(x) > 0$ with the

*Orlicz norm*:

$|| x ||^{0}_{\phi}$ = sup $\{\int _{\Omega } |x(t)y(t)|d\mu(t) : \int _{\Omega } \phi^{^{*}}(|y(t)|)d\mu(t) ≤ 1 \}$

or the

*Luxemburg-Nakano norm*:

$|| x ||_{\phi}$ = inf $\{\lambda > 0 : \int _{\Omega } \phi(|x(t)|/\lambda)d\mu(t) ≤ 1 \}$

Orlicz spaces $L^{\phi}$ are a natural generalization of $L^{p}$ spaces. They have very rich topological and geometrical structure; they may possess peculiar properties that do not occur in ordinary $L^{p}$ spaces.

Orlicz's ideas have inspired the research of many mathematicians. In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc. (cf. monographs on Orlicz spaces: M A Krasnoselskii and Ya B Rutickii,

*Convex Functions and Orlicz Spaces*(Groningen 1961), J Lindenstrauss and L Tzafriri,

*Classical Banach Spaces*I, II (Springer 1977, 1979), C Wu and T Wang,

*Orlicz Spaces and their Applications*, (Harbin 1983 - Chinese), A C Zaanen,

*Riesz Spaces*II, (North-Holland 1983), C Wu, T Wang, S Chen and Y Wang,

*Theory of Geometry of Orlicz Spaces*(Harbin 1986 - Chinese), L Maligranda,

*Orlicz Spaces and Interpolation*, (Campinas 1989), M M Rao and Z D Ren,

*Theory of Orlicz Spaces*(Marcel Dekker 1991) and S Chen,

*Geometry of Orlicz Spaces*(Dissertationes Math. 356, 1996).

The term

*Orlicz spaces*appeared in the sixties in the Mathematics Subject Classification index of the American Mathematical Society in Section

**4635**, which is now

**46E30**,

*Spaces of measurable functions (Lp -spaces, Orlicz spaces, etc.*).

To emphasize the importance of Orlicz spaces in a jocular way, Professor Orlicz used to say that when he was occasionally asked:

Why are Orlicz spaces "better" than Lp spaces?he liked to answer:

Tell me first why $L^{p}$ spaces are "better" than $L^{2}$?Here is another anecdote in connection with Orlicz spaces:

Professor Orlicz had a small apartment and he once applied to the city administration for a bigger one. The answer of an employee was:

Your apartment is really small but we cannot accept your claim since we know that you have your own spaces !Orlicz's name is associated not only with the Orlicz spaces but also with the Orlicz-Pettis theorem, Orlicz property, Orlicz theorem on unconditional convergence in $L^{p}$, Mazur-Orlicz bounded consistency theorem, Mazur-Orlicz theorem on inequalities, Mazur-Orlicz theorem on uniform boundedness in $F$-spaces, Orlicz category theorem, Orlicz interpolation theorem, Orlicz norm, Orlicz function, convexity in the sense of Orlicz, $F$-norm of Mazur-Orlicz, Drewnowski-Orlicz theorem on representation of orthogonal additive functionals and modulars, Orlicz theorem on Weyl multipliers, Matuszewska-Orlicz indices, Hardy-Orlicz spaces, Marcinkiewicz-Orlicz spaces, Musielak-Orlicz spaces, Orlicz-Sobolev spaces and Orlicz-Bochner spaces.

For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.

The Orlicz theorem on unconditional convergence in $L^{p}$is:

If $1 ≤ p < ∞$ and $\sum f_{n}$ is unconditionally convergent in $L^{p}$, then ∑ || $f_{n}$|| max( class="math">$p$,2) < ∞.

In 1988, on the occasion of Orlicz's 85-th birthday, Polish Scientific Publishers (PWN) published his

*Collected Papers*[3] in two volumes with a total of 1754 pages, reproducing 141 his articles from 1926-1985.

Kuratowski [1, p. 40] has written about creation of the Polish School of Mathematics stating that:

...functional analysis owes its magnificient development to Banach and his students, especially to Mazur, Orlicz and Schauder.Orlicz's scientific achievements are presented in detail in the papers by Maligranda-Matuszewska [7], Maligranda-Wnuk [11] and Maligranda [6]. Paper [11] contains a complete list of Orlicz's publications (171 papers and 3 books).

In the late seventies Orlicz started to collect information about mathematicians from Lvov and he was planning to write a book on the

*History of the Lvov School of Mathematics*(he published only two articles:

*The Lvov School of Mathematics between the Wars*, Wiadom. Mat. 23(1981), 222-231 and

*Achievements of Polish Mathematicians in the Domain of Functional Analysis in the Years*1919 - 1951, and biographies of S Banach, S Kaczmarz, A Lomnicki, S Mazur, J P Schauder). It is a real pity that he did not finish this project.

Orlicz died on 9 August, 1990 in Poznań when correcting the galley proofs of his last paper accepted for publication in

*Mathematica Japonica*.

Three conferences were organized in the memory of Władysław Orlicz:

*Orlicz Memorial Conference*(March 21 - 23, 1991) by the University of Mississippi in Oxford, USA;

*Function Spaces V*(28 August to 2 September, 1998) by the University of A. Mickiewicz in Poznań, Poland (paper [10] appeared in the proceedings of this conference);

*Scientific Session in the Memory of Professor Władysław Orlicz*(September 27-29, 2000) by the University of A. Mickiewicz and the Institute of Mathematics of the Polish Academy of Sciences in Bedlewo, Poland (the proceedings include paper [6] with over forty photos of Orlicz).

### References (show)

- K Kuratowski,
*A half century of Polish mathematics*(Warsaw, 1980). - H Steinhaus,
*Between spirit and matter mediate mathematics*(Polish) (Warsaw-Wrocław, 2000). *Wladyslaw Orlicz Collected Papers*I, II (Warsaw, 1988).- J Albrycht, Speech delivered at the graduation ceremony on honorary membership of W. Orlicz (Polish),
*Wiadom. Mat.***18**(1974), 200-204. - W Jankowski, On the scientific activity of Professor W. Orlicz (Polish),
*Wiadom. Mat.***22**(1980), 275-279. - L Maligranda, Wladyslaw Orlicz (1903-1990) - his life and contribution to mathematics (Polish), In: Wladyslaw Orlicz (1903-1990) - Founder of the Poznan School of Mathematics,
*Poznan*2001, 33-79. - L Maligranda and W Matuszewska, A survey of Wladyslaw Orlicz's scientific work, In:
*Wladyslaw Orlicz Collected Papers*, (Warsaw 1988) xv-liv. - L Maligranda and L E Persson, Wladyslaw Orlicz (1903-1990),
*Notices Amer. Math. Soc.***38**(1991), 21. - L Maligranda and W Wnuk, Wladyslaw Orlicz(1903-1990) (Polish),
*Nauka Polska***3**(1992), 187-193. - L Maligranda and W Wnuk, Wladyslaw Orlicz: his life and contributions to mathematics, In:
*Function Spaces*(New York-Basel 2000), 23-29. - L Maligranda and W Wnuk, Wladyslaw Orlicz (1903-1990) (Polish),
*Wiadom. Mat.***36**(2000), 85-147. - W Matuszewska, Wladyslaw Orlicz - A review of his scientific work,
*Commentationes Math. Tomus Specialis in Honorem Ladislai Orlicz*,*I*, (1978), 1-17. - W Matuszewska, Wladyslaw Orlicz, In:
*Wladyslaw Orlicz Collected Papers,*(Warsaw 1988), xi-xiv. - Z Semadeni, On W. Orlicz papers in functional analysis (Polish),
*Wiadom.Mat.***18**(1974), 191-199. - Wladyslaw Orlicz: 24 May 1903 - 9 August 1990,
*Studia Math.***97**(1990), No. 2, ii-iv.

### Additional Resources (show)

Other pages about Władysław Orlicz:

Other websites about Władysław Orlicz:

### Cross-references (show)

Written by Lech Maligranda, Lulea University of Technology, Sweden.

Last Update May 2001

Last Update May 2001