Foundations of Analysis

Analysis is the rigorous foundation underneath calculus, and this book covers the essential pillars: inequalities, series convergence, multiple integrals, and functional analysis.

The triangle inequality chapter is thorough. It covers the geometric idea, the real number case, the Cauchy-Schwarz inequality, extensions to vectors, complex numbers, metric spaces, normed spaces, and the integral form. The reverse triangle inequality and applications round out the treatment.

Series convergence is handled via D’Alembert’s ratio test, Cauchy’s root test, Raabe’s test, the integral test, and comparison tests. Each test comes with worked examples and clear statements of when it applies.

The Dirichlet theorem chapter covers the Gamma function (Euler and Weierstrass definitions), the Beta function, Dirichlet’s theorem on multiple integrals, Liouville’s extension, and the Dirichlet distribution.

The functional analysis introduction covers vector spaces, normed spaces, Banach spaces, inner product spaces, Hilbert spaces, bounded linear operators, and the fundamental theorems (Hahn-Banach, open mapping, closed graph, uniform boundedness).

Topics Covered

• Triangle inequality in all settings (real, complex, metric, normed)
• Complete suite of convergence tests for series
• Gamma and Beta functions with key identities
• Dirichlet’s theorem and Liouville’s extension
• Banach spaces and Hilbert spaces
• The four fundamental theorems of functional analysis