Definition:Algebraic Dual

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This page is about Algebraic Dual in the context of Linear Algebra. For other uses, see Dual.


Let $\struct {R, +, \times}$ be a commutative ring.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

Let $\struct {R, +_R, \circ}_R$ denote the $R$-module $R$.

The $R$-module $\map {\LL_R} {G, R}$ of all linear forms on $G$ is usually denoted $G^*$ and is called the algebraic dual of $G$.

Double Dual

The double dual $G^{**}$ of $G$ is the dual of its dual $G^*$.

Also see

  • Results about algebraic duals can be found here.