Linear Differential Equations and the Sturm-Liouville Problem. The ``Crypto-Integral'' Equations. The Equation of Vibrating Membranes. The Idea of Infinite Dimension. The Crucial Years and the Definition of Hilbert Space. Duality and the Definition of Normed Spaces. Spectral Theory after 1900. Locally Convex Spaces and the Theory of Distributions. Applications of Functional Analysis to Differential and Partial Differential Equations. References. Author and Subject Index.