The paper surveys the gradual rise and initial adoption in the field of analysis of the concept of vector spaces of functions, in the period from ca. 1880 to ca. 1930. Some early formalistic approaches notwithstanding, the treatment of linear problems for an infinite number of dimensions long bore the mark of the analogy with finite-number of dimensions situations, this at the time involving the use of determinants. The paper examines how the Fredholm equation, on the one hand, and particularly Hilbert's contribution on the matter, and the emergence of topological concepts on the other hand, came to show, through a succession of generalisations, the requirement for an axiomatic approach.