Variational Principles of Continuum Mechanics

Abstract

Let V be a linear vector space over the complex numbers, such that for any pair of vectors f, g, ɛ V there exists a complex number <f,g> and the maps VxV→C called the inner (or scalar) product ·{f,g} →<f,g> satisfies the following axioms

• i) $\left\langle {f,g} \right\rangle = \left\langle {\overline {g,f} } \right.$ (bar denotes complex conjugate)

• ii) for any ${{a}_{1}},{{a}_{2}} \in C,{{f}_{1}},{{f}_{2}},g \in V$.

• iii) 〈f,f〉 is a nonnegative real number (for any f ∈ V) and 〈f,f〉 $= \Leftrightarrow f = \emptyset .$.

• iv) It follows from i) and iii) that $\left\langle {af,g} \right\rangle = a\left\langle {f,g} \right\rangle = \left\langle {f,\bar{a}g} \right\rangle .$.