Definition:Algebraic Dual

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

Definition

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

• Set of Linear Transformations over Commutative Ring forms Submodule of Module of All Mappings

• Definition:Dual Vector Space
• Definition:Normed Dual Space

• Results about algebraic duals can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations