An elementary introduction to general topology, Lebesgue integration theory, and functional analysis presented in a unified manner based on real analysis and linear algebra. Applications include differential and integral equations, optimization, and numerical methods. Starting from well-known concepts such as metric, convergence, vector space, or scalar product, topological, specifically pseudometric spaces, are discussed along with their fundamental properties as they are used in modern mathematics. Following a classical presentation of the abstract Lebesgue integration of functions of one variable, five principles of linear functional analysis are introduced, along with the Banach-Hahn-Mazur separation theorems and investigations into extreme points, duality, and Hilbert space operators. The text includes over 150 examples and 428 exercises with complete solutions.