Introduction to the Variational Formulation in Mechanics

BElements of Real and Functional Analysis

B.1 Introduction

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 ( and ).

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 , in general, . Thus, it is necessary to introduce further ...