The first volume of a book on mathematical tools for solving differential equations from physics, which has five parts: semi-normed spaces, distribution, Lebesgue and Sobolev spaces, traces, partial differential equations (PDEs). This book presents the main properties of semi-normed or normed spaces useful for solving partial differential equations from physics. As well as the Banach, Hilbert and Frchet spaces, the focus is on Neumann spaces, i.e. sequentially complete spaces. Differential calculus is extended to semi-normed spaces in preparation for the construction of Lebesgue and Sobolev distribution spaces with real or vector values given in Volumes 2 and 3.