Dimension of Double Dual

Theorem

Let $R$ be a commutative ring with unity.

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

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

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

Proof

By definition, the double dual of $G$ is the algebraic dual of the algebraic dual $G^*$ of $G$.

From Dimension of Algebraic Dual:

$\map \dim {G^**} = \map \dim {G^*}$

Also from Dimension of Algebraic Dual::

$\map \dim {G^*} = \map \dim G$

Hence the result.

$\blacksquare$

Sources

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