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This textbook gives an introduction to optimization tools which arise around the Weierstrass theorem about the minimum of a lower semicontinuous function. Starting from a Euclidean space, it moves further into the infinite dimensional setting towards the direct variational method, going through differentiation and introducing relevant background information on the way. Exercises accompany the text and include observations, remarks, and examples that help understand the presented material. Although some basic knowledge of functional analysis is assumed, covering Hilbert and Banach spaces and the Lebesgue integration, the required background material is covered throughout the text, and literature suggestions are provided. For less experienced readers, a summary of some optimization techniques is also included. The book will appeal to both students and instructors in specialized courses on optimization, wishing to learn more about variational methods. Preface The Weierstrass Theorem the Origin of Optimization Introductory Remarks Lower Semicontinuity and the Weierstrass Theorem Applications to Minimization Problems Applications to Global Invertibility Applications to Algebraic Equations The Lagrange Multiplier Rule Some Basics from Functional Analysis and Function Spaces On the Convexity with Some Revision from the Calculus On the Weak Convergence Niemytskij Operator and the Krasnosel'skii-Type Theorem On the Space C[0,1] On Absolutely Continuous Functions On the Spaces H1(0,1) and H01(0,1) Definition and Basic Properties Embeddings of the Space H01(0,1) The Space H-1(0,1) On the du Bois-Reymond Lemma and the Regularity Results Differentiation in Infinite-Dimensional Spaces The Gâteaux Variation and Its Computation On the Fermat Rule The Gâteaux Derivative The Fréchet Derivative On the Differentiability of Maps Between Banach Spaces More on the Convexity On the Weierstrass Theorem in Infinite-Dimensional Spaces Direct Variational Method Some Remarks on Approximation Problems Minimization of the Classical Euler Action Functional Applications to Control Problems Applications to Second-Order Dirichlet Problems On the Best Constant in the Poincaré Inequality On Some Abstract Formulation of the Direct Method Applications to Multiple Integrals Instead of an Introduction On the Space C( Ω) Sobolev Spaces Some Applications to Integral Functionals On the Dirichlet Problem References Index