Presented, with some minor differences, as a PhD thesis on 10 March 1944 in secret to the University of Lwów, during the terror of the German occupation.

Scalar and vector measurable functions; sequences of linear operators; the Denjoy integral; differentiation of vector functions; differential equations and equations with vector functions; two norm spaces and two norm algebras and their applications in summability theory; analytic functions; and applications of functional analysis to classical problems of mathematical analysis.

... presents the basic theory of Banach spaces, rich material on non-locally convex and non-measurable linearly topological spaces and numerous examples of applications of functional analysis.

The scientific output of Andrzej Alexiewicz indicates his immense creativity and intellectual power. It reveals his ability to understand the core of a problem, grasp the inter-relations and dependencies between different notions, and solve natural, well-motivated problems in a clear and elegant way.

Art played a special role in the life of Andrzej Alexiewicz. His greatest passions were mathematics, painting and music. He painted whenever he could. He would take his easel when going for walks outside Poznań, on holidays at the seaside or in the mountains, where he worked in unique places that he found. He often made sketches to fix colours and landscapes so that he was able to remember them later in his university office which he had partially converted into a studio. In his house, pictures on walls often changed but the most important place was always occupied by a beautifully framed large oil-landscape from Czarnohora, painted in 1935. The painting presented the waterfall "Huk" in Kosmacz, one of the favourite places of professor Alexiewicz. The professor was especially fascinated with the continually changing colours of the sea and the sky, with conversations with curious tourists as he set up his easel on beaches in Mielno, Dziwnow or Kolobrzeg, and on family walks at sunrises and sunsets by the Baltic sea. He painted numerous views of the sky and sea-waves, often presented in abstract forms. He also fixed his vision of the world in many mountainous landscapes from the Eastern Carpathians, the Tatras, the Pieniny and California, and also in numerous pictures painted directly out of windows of his studio. In those last landscapes he showed unknown Poznań, its plants, the changeable lights and colours of seasons of the year in the city. ... Professor Andrzej Alexiewicz had a discerning musical taste. He appreciated classical symphonic music, especially compositions of Johann Sebastian Bach, whom he held to be the greatest composer. He often went to concerts with the whole family to the University hall. The professor especially enjoyed the famous film "Amadeus" by Milo Forman. He saw it many times, encouraged all his friends to see it and later asked about the number of visits in the cinema. After his seminars and lectures he often invited students to listen to music together in his office. He also listened to music while painting. Sometimes after an especially successful symphonic concert he would go to his studio, where he, touched by music, started to paint. He was always interested in the opinions of guests about his paintings. He was very glad when his studio was visited by painters and he could hold discussions with them.