For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s *That’s Maths* column in *The Irish Times* (TM063, or search for “thatsmaths” at irishtimes.com).

Zygmunt Janeszewski, who had been awarded a doctorate by the Sorbonne in 1911, had a vision of Polish mathematical greatness and devised a programme for its achievement. He advocated that Polish mathematicians should specialise in a few clearly-defined fields, rather than trying to cover too many areas. This would ensure common interests and foster a culture of collaboration.

A plan like this can succeed only if there are talented mathematicians to carry it out. Fortunately, while there was no strong tradition of excellence, several brilliant Polish mathematicians emerged around that time. The leading lights were Hugo Steinhaus, with a doctorate from Göttingen – then the Mecca of mathematics – and Stefan Banach, who would become the greatest Polish mathematician.

**The Lvov School**

Diverse contributions to mathematics were made by the Lvov School, earning it world-wide admiration. Names like Banach, Sierpiński, Kac and Ulam occur frequently in modern textbooks. They were concerned with fundamental aspects of mathematics: axiomatic foundations of set theory, functions of a real variable, the nature of general function spaces and the concept of measure.

The year 1932 saw the publication of Banach’s monograph on normed linear spaces, *Theory of Linear Operations*. It contained many powerful results. His genius was to combine different areas of mathematics. He treated functions as points in an abstract space which was linear, with a concept of distance and an absence of “gaps”: a complete, normed linear space, a fusion of algebra, analysis and topology. It proved eminently suitable for the development of the field called functional analysis,

Banach’s monograph had a major influence and his notation and terminology were widely adopted. His spaces quickly became known as Banach spaces and they have played a central role in functional analysis ever since. They also served as a foundation for quantum mechanics. Banach’s monograph established the international importance of the Lvov School.

**Collaborative Mathematics**

Many of the mathematical breakthroughs in Lvov resulted from collaborations and the majority of the publications are work of two or more authors. The mathematicians used to meet regularly in cafés, discussing mathematics late into the night.

Their favourite haunt was *The Scottish Café*, undoubtedly the most mathematically productive café of all time. The table-tops were of white marble on which mathematics could be easily written (and erased).

Later, Banach’s wife bought a large notebook for the group – the famous Scottish Book – in which problems and solutions were recorded. This was kept in the café and was available to any mathematicians who visited. Ultimately, it contained about 200 problems, many of which remain open to this day.

One problem caused a media sensation when it was finally solved. Stanisław Mazur had offered a live goose for a solution and, in 1973, the Swedish mathematician Per Enflo travelled to Warsaw to collect his prize from Mazur for solving “the Goose problem”.

The Scottish Café exemplified the synergy and camaraderie that pervaded Polish mathematics in the inter-war years. World War II changed everything. Polish culture was systematically eradicated. Steinhaus managed to escape execution by assuming a false identity.

Banach survived to witness the defeat of Nazism but died shortly afterwards. Today, the city is Lviv, a major centre of culture in western Ukraine. The Golden Age of the Lvov School has passed into history.

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