After studying composition and music theory in Oslo with Johan Svendsen and Ludvig Mathias Lindeman, he went to the Leipzig Conservatory in Germany for two years.
On his return to Norway in 1889, he worked as a music critic for several newspapers and successfully premiered his cantata, Hvæm er du med de tusene navne (Who are you with a thousand names).
However, in 1890, he left Norway and was to live for the next thirteen years in Leipzig and Berlin, where he became a friend of the Italian composer and pianist Ferruccio Busoni whose musical philosophy he shared.
[3] Borgstrøm returned to Norway in 1903, the year in which his symphonic poem for piano and orchestra, Hamlet, premiered to great acclaim in Oslo.
Although, he is primarily known for his symphonic works, written in a late Romantic style with influences of Expressionism, he also composed 45 songs, most notably "Svalerne" (The Swallows), "Rød valmue" (Red Poppy), and "Frossen skog" (Frozen Forest), as well as two operas, neither of which was performed in his lifetime.