B: Elements of Real and Functional Analysis
Edgardo O. Taroco, Pablo Javier Blanco, Raúl A. Feijóo · 2019
In the previous appendix we introduced a series of concepts and definitions which are required from Chapter 3 onwards.In particular, the definitions of the translation of a subpsace, of a cone, the concept of dual space, orthogonal complement, and conjugate cone, among others, were useful in the conception of the Principle of Virtual Power and the Principle of Complementary Virtual Power.The same happens with the concept of the adjoint operator, which in the mechanical realm corresponds to the concept of the equilibrium operator (𝒟 and 𝒟 * ).However, in the previous appendix as well as, to some extent, in Chapter 3, we employed a purely algebraic vision that, for the case of finite-dimensional vector spaces, was enough to scrutinize the underlying theoretical concepts.Strictly, this is not enough if we are dealing with infinite-dimensional spaces (as is the case of the spaces usually employed in the mechanics of continua) because, for example, if is a vector subspace of X, in general, ( ⟂ ) ⟂ ⊂ .Thus, it is necessary to introduce further concepts such as those of open and closed sets, convergence and continuity, among others, which are associated with the topological structure that is required by such vector spaces for the mathematical treatment to be correct.In order to do this, in this appendix we will briefly address some concepts of the real number system for which we consider the usual addition and multiplication operations, as well as the natural ordering of the real numbers (if x ≤ y then x + z ≤ z + y, ∀z ∈ ℝ; if x ≥ 0 and y ≥ 0 then xy ≥ 0 and, therefore, "≤" is a complete ordering!).If x ∈ ℝ, the absolute value of x, denoted by |x|, is defined byConsider x, y ∈ ℝ, then it is verified the triangle inequality isAlso, observe that, for x, y ∈ ℝ, the absolute value satisfies |x -y| ≥ 0, |x -y| = 0 if and only if x = y and |x -y| = |y -x|.Moreover, the following difference between absolute values is also verified, establishing, for x, y ∈ ℝ, that ||x| -|y|| ≤ |x -y|. (B.3)Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications, First Edition.