Dimension of Algebraic Dual

Theorem

Let $R$ be a commutative ring with unity.

Let $G$ be an $n$-dimensional $R$-module.

Let $G^*$ be the algebraic dual of $G$.

Then $G^*$ is also $n$-dimensional.

Dimension of Double Dual

Let $G^{**}$ be the double dual of $G$.

Then $G^{**}$ is also $n$-dimensional.

Proof

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

From Dimension of $R$-Module $R$ is $1$ we have that the dimension of $\struct {R, +_R, \circ}_R$ is $1$.

By definition, the algebraic dual of $G$ is the $R$-module $\map {\LL_R} {G, R}$ of all linear forms on $G$.

From Dimension of Set of Linear Transformations the dimension of $G^*$ is the product of the dimension of $G$ with the dimension of $\struct {R, +_R, \circ}_R$.

That is:

$\map \dim {G^*} = n \times 1$

Hence the result.

$\blacksquare$

Sources

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