After the collapse of Austria, Schauder joined a new Polish army which was being organised in France. He returned to Poland with this army in 1919, by this time the country of Poland had been reestablished and it was to a Polish Lvov that Schauder went. He entered the Jan Kazimierz University in Lvov and worked for his doctorate under Steinhaus.
After obtaining his doctorate in 1923 with a thesis The theory of surface measure, he taught both in a secondary school and worked for an insurance firm. In 1927 he wrote the paper Contributions to the theory of continuous mappings on function spaces and was allowed to teach at the university on the strength of this work. His first courses were given at the University in Lvov during 1928-29 but he continued also with his position as a secondary school teacher. As Ulam writes in :-
Although Lvov was a remarkable centre for mathematics, the number of professors both at the Institute and at the University was extremely limited and their salaries were very small. ... Schauder had to teach in high school in order to supplement a meager income as lecturer ...Schauder married Emilia Löwenthal in 1929. She also came from a Jewish family although her grandfather had been expelled from the Jewish community on the grounds that he was an atheist. They would have one daughter Eva Schauder.
Brouwer published his fixed point theorems in 1911 for finite dimensional spaces. Schauder published fixed point theorems for Banach spaces in 1930. In 1932 he was awarded a Rockefeller scholarship which enabled him to spend part of 1932-33 Leipzig. Still financed by the scholarship in May 1933 he moved to Paris to work with Hadamard.
While Schauder was in Paris he collaborated with J Leray and their joint work led to a paper Topologie et équations fonctionelles published in the Annales scientifiques de l'École Normale Supérieure. This 1934 paper on topology and partial differential equations is of major importance :-
In this paper what is now known as Leray-Schauder degree (a homotopy invariant) is defined. This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.His last work was to generalise results of Courant, Friedrichs and Lewy on hyperbolic partial differential equations. In 1938 he received the Grand Prix Internationaux de Mathématiques Malaxa (jointly with Leray) for the work of the 1934 paper. In fact by the time Schauder received this prize his final publication (1937) had appeared in print. His short career was about to come to an end with the start of World War II but, despite his publications spanning only 10 years, he had written 33 works. Kuratowski, in , sums up Schauder's main mathematical contributions:-
Schauder's main achievement consists in transferring some topological notions and theorems to Banach spaces (the fixed point theorem, invariance of domain, the concept of index). In particular, Schauder's formulation of a fixed point theorem originated a new, extremely fruitful method in the theory of differential equations, known as Schauder's method ...Forster, in , writes:-
Schauder's fixed point theorem and his skillful use of function space techniques to analyse elliptic and hyperbolic partial differential equations are contributions of lasting quality. Existence proofs for complicated nonlinear problems using his fixed point theorem have become standard. The topological method developed in the 1934 Leray-Schauder paper ... is now utilised not only to obtain qualitative results but also to solve problems numerically on computers.In 1939, at the beginning of the World War II, Soviet troops occupied Lvov. Schauder was treated well by the new Soviet administration. He was appointed to professor at the university, now renamed the Ivan Franko University. In June 1941 the German army entered Lvov and a systematic extermination of Jews began. Schauder sent pleas for help to Hopf and Heisenberg saying he had many important results but no paper to write them on.
There are two versions of how he died and it is impossible to tell which is correct. One version states that he was betrayed to the Gestapo who then arrested him and, like many Jews, he was never seen again. The second version of his death (thought by Forster the author of  to be more likely) is the he was shot by the Gestapo in September 1943 in one of their regular searches for Jews.
Schauder's wife Emilia was hidden in Lvov by the Polish resistance for some time after her husbands death. She was hidden in the sewers of Lvov with her daughter Eva. However, eventually she surrendered to the Germans and was sent to Lublin concentration camp where she died. Eva, Schauder's daughter, survived until the end of the war when she went to Italy to live with Schauder's brother who lived there.
We should comment on the picture of Schauder. Ulam  explains that Leray, the French mathematician with whom Schauder collaborated, wrote to him several years after the end of World War II:-
Leray wanted to have a photograph of [Schauder] for himself and for Schauder's daughter who survived the war and lives in Italy. But he could not find any in Poland or anywhere and he wrote to me asking whether I maght have a snapshot. Some months after Johnny von Neumann's death I was looking at some of the books in his library and a group photo of the participants in the  Moscow conference fell out. Schauder was there, as were Aleksandrov, Lefschetz, Borsuk, and some dozen other topologists. I sent this photograph to Leray.
Article by: J J O'Connor and E F Robertson