 Another immediate consequence of the concrete approach is duality for finite-dimensional vector spaces over \(F\). To every vector space \(V\), of any dimension, corresponds its dual space \(V^*\) comprised of the functionals on \(V\), defined as the linear transformations \(f: V\rightarrow F\), viewing the field \(F\) as the one-dimensional vector space. The functionals form a vector space under coordinatewise addition \((f+g)(u) = f(u)+g(u)\) and multiplication \((xf)(u) = x(f(u))\) by any scalar \(x\) in \(F\), and we take \(V^*\) to be that space. This operation on vector spaces extends to the linear transformations \(f: U\rightarrow V\) as \(f^* : V^*\rightarrow U^*\) defined such that \(f\) maps each functional \(g: V\rightarrow F\) to \(g\cdot f: U\rightarrow F\). Repeating this operation produces a vector space that, in the finite-dimensional case, is isomorphic to \(V\), that is, \(V \cong V^{**}\), making the operation an involution. The essence of duality for finite-dimensional vector spaces resides in its involutary nature along with the reversal of the linear transformations.