Editing Dual basis (section)

==A categorical and algebraic construction of the dual space==

Another way to introduce the dual space of a vector space ([[Module (mathematics)|module]]) is by introducing it in a categorical sense. To do this, let <math>A</math> be a module defined over the ring <math>R</math> (that is, <math>A</math> is an object in the category <math>R\text{-}\mathbf{Mod}</math>). Then we define the dual space of <math>A</math>, denoted <math>A^{\ast}</math>, to be <math>\text{Hom}_R(A,R)</math>, the module formed of all <math>R</math>-linear module homomorphisms from <math>A</math> into <math>R</math>. Note then that we may define a dual to the dual, referred to as the double dual of <math>A</math>, written as <math>A^{\ast\ast}</math>, and defined as <math>\text{Hom}_R(A^{\ast},R)</math>.

To formally construct a basis for the dual space, we shall now restrict our view to the case where <math>F</math> is a finite-dimensional free (left) <math>R</math>-module, where <math>R</math> is a ring with unity. Then, we assume that the set <math>X</math> is a basis for <math>F</math>. From here, we define the Kronecker Delta function <math>\delta_{xy}</math> over the basis <math>X</math> by <math>\delta_{xy}=1</math> if <math>x=y</math> and <math>\delta_{xy}=0</math> if <math>x\ne y</math>. Then the set <math> S = \lbrace f_x:F \to R \; | \; f_x(y)=\delta_{xy} \rbrace </math> describes a linearly independent set with each <math>f_x \in \text{Hom}_R(F,R)</math>. Since <math>F</math> is finite-dimensional, the basis <math>X</math> is of finite cardinality. Then, the set <math> S </math> is a basis to <math>F^\ast</math> and <math>F^\ast</math> is a free (right) <math>R</math>-module.