Dual Vector Space

Let be a Vector Space over Field
The dual is , the space of all Linear functions from to

Proposition

This is well defined

Proposition

There is a natural injective homomorphism between and .
This is actually a Natural Transformation.

Proof

Let
Define a homomorphism
by for all
and denote this homomorphism by

Then is a homomorphism
Now just check injectivity.

Proposition

When is finite dimensional, it is isomorphic to .

Proof

Use the natural injective homomorphism
Now note that and
Thus and, by picking a basis, we get that is surjective.