Page 151 - DMTH502_LINEAR_ALGEBRA

Unit 11: The Double Dual

Notes
If is a vector in V, then includes a linear functional L on V defined by
*
V
f
L f f , in *. ...(1)
The fact that L is linear is just a reformulation of the definition of linear operations in V :
*
L cf g cf g

cf g

cf g

cL f L g . ...(2)

If V is finite-dimensional and 0, then L 0; in other words, there exists a linear functional f
such that f 0. The proof is very simple. Choose an ordered basis  = 1 ,..., n for V such
that = and let f be the linear functional which assigns to each vector in V its first coordinate
1
in the ordered basis  .

Theorem 1: Let V be a finite-dimensional vector space over the field F. For each vector in V
define

L f f , in *.
f
V
The mapping L is then an isomorphism of V onto V**.
Proof: We showed that for each the function L is linear. Suppose and are in V and c is in

F, and let c . Then for each f in V*.
L f f

f c

cf f

and so cL f L f

L CL L

This shows that the mapping L is a linear transformation from V into V**. This
transformation is non-singular; for, according to the remarks above L 0 if and only if 0.
Now L is a non-singular linear transformation from V into V**, and since

dim V** = dim V* = dim V ...(3)
Therefore this transformation is invertible, and is therefore an isomorphism of V onto V**.

Corollary: Let V be a finite-dimensional vector space over the field F. If L is a linear functional
on the dual space V* of V, then there is a unique vector in V such that

L f f ...(4)
for every f in V*.

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