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Math Help - vector space and its dual space

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

    This is like taking the functional f and projecting it on W^*. Or, if you want to put it another way : f is the image, under a (non-canonical) isomorphism \pi : V\rightarrow V^*, of some vector v; first projecting v on W and then sending, via \pi, the resulting projection to W^*, will give the restriction of f to 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.
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  3. #3
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    Quote Originally Posted by aman_cc View Post
    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 i:W\subset V is the inclusion map of W into V, then 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**).
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    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
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