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.