Duality

In physics, there is a tendency to believe that dual vector spaces \(V^* = {{\,\textrm{Hom}\,}}(V, \mathbb {K})\) are unnecessary and superfluous. This can be fallacious and cause unnecessary obstacles to understanding crucial aspects of linear algebra related to theoretical physics. Duality is also essential for understanding tensor formalism, the subject of the following chapter. A particular difficulty with the dual space is that we have to know precisely in what situations one must and in what situations one need not pay close attention to it. Even in this case, the elements of \(V^*\) , which appear as elements of V, have a different transformation behavior from the standard elements in V. This is why our intention in this section is also to help avoid such confusion. To clarify this situation, we have to proceed slowly and repeat, when necessary, some well-known facts. So we will distinguish precisely the formalism corresponding to an abstract vector space (without an inner product) from a vector space with a scalar product. We believe that this section is necessary to understand tensor formalism. This makes it indispensable for a good under- standing of special and general relativity and relativistic field theory.