# Thread: vector space and its dual space

1. ## vector space and its dual space

Q1. Let V be a vector space. V* be it's dual space i.e. set of all linear functions on V. W is a sub-space of V. W* be dual space of W i.e. set of all linear functions on W. Am I correct in saying that V* is a sub-space of W*? Because every f in V* will also be in W*.

I know I'm wrong but just not getting this.

Q2. Let V, V*, V** be vector space, it's dual space and it's double dual space. Can anyone help me construct a homomorphism from V* into V**, kernel of the homomorphism = A(W) annihilator of a sub-space of W, thus subspace of V*. I would want to prove that this homomorphism is onto W**. And thus derive the result that V*/A(W) is isomorphic to W** and hence W.

Thanks for reading and any help.

2. #1) This is incorrect, but you are not completely wrong. A linear functional on $\displaystyle V$, i.e. an element $\displaystyle f$ of $\displaystyle V^*$, is a function $\displaystyle V\rightarrow F$. You are right in saying that $\displaystyle f$ is defined on $\displaystyle W$, but it is also possibly defined elsewhere. The restriction of $\displaystyle f$ to $\displaystyle W$ is an element of $\displaystyle W^*$.

This is like taking the functional $\displaystyle f$ and projecting it on $\displaystyle W^*$. Or, if you want to put it another way : $\displaystyle f$ is the image, under a (non-canonical) isomorphism $\displaystyle \pi : V\rightarrow V^*$, of some vector $\displaystyle v$; first projecting $\displaystyle v$ on $\displaystyle W$ and then sending, via $\displaystyle \pi$, the resulting projection to $\displaystyle W^*$, will give the restriction of $\displaystyle f$ to $\displaystyle W^*$.

I haven't checked the details but I'm quite sure the above is right - you might want to check, I could just be high on coffee.

3. Originally Posted by aman_cc
Q1. Let V be a vector space. V* be it's dual space i.e. set of all linear functions on V. W is a sub-space of V. W* be dual space of W i.e. set of all linear functions on W. Am I correct in saying that V* is a sub-space of W*? Because every f in V* will also be in W*.
[This overlaps with Bruno J's comment above.]
It is not true that V* is a subspace of W*. In fact, the dual of an inclusion map is a projection. So if $\displaystyle i:W\subset V$ is the inclusion map of W into V, then $\displaystyle i^*:V^*\to W^*$ is the projection from V* onto W*. The kernel of i* is A(W), and so i* exhibits an isomorphism between W* and V*/A(W).

It's also true that the dual of a projection is an inclusion. So the second dual map i** is an inclusion of W** in V** (which agrees with the original inclusion W⊂V under the canonical identification of V with V**).

4. Thanks guys !! I guess I will take time to digest all you guys wrote. Can I request help on Q2 in the post as well.(greedy me!!)

Also to understand what you wrote - should I be knowing about things like inclusion map, projection and non-canonical? I haven't really read that part of theory.

Thanks