This communication investigates how the notion of vector space of functions gradually became vital in analysis, roughly between 1880 and 1930. We will show how, in spite of some early formal approaches, linear problems in infinite dimension remained long dependent on the analogy with finite dimensional theory, which was still dominated by the theory of determinants. Subsequently, we will see how, on one side, the study of the Fredohlm equation, especially Hilbert's work, and on the other side topological considerations, led, through successive processes of generalisation to the need for an axiomatical approach.