Normed space and its double dual

  • Thread starter dmuthuk
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Hi, I am trying to sort out a few things about the dual and double dual of a normed space which has got me a little confused.

Given a normed space [tex]X[/tex] over [tex]\mathbb{R}[/tex], if [tex]Y[/tex] is a subspace of [tex]X[/tex], what is the relationship between [tex]Y^*[/tex] and [tex]X^*[/tex]? Can [tex]Y^*[/tex] be identified with some subspace of [tex]X^*[/tex]?

Also, we have the natural embedding of [tex]X[/tex] into [tex]X^{**}[/tex] given by the map [tex]x\mapsto\hat{x}[/tex] where [tex]\hat{x}:X^*\to\mathbb{R}[/tex] is evaluation at [tex]x[/tex]. How do we use this to define the natural embedding of [tex]Y[/tex] into [tex]Y^{**}[/tex]? Here is my idea: We want to send [tex]y[/tex] to [tex]\hat{y}_Y:Y^*\to\mathbb{R}[/tex]. So, we define [tex]\hat{y}_Y[/tex] as follows. For [tex]y^*\in Y^*[/tex], choose an arbitrary extension [tex]x^*\in X^*[/tex] (we can use Hahn-Banach here). Then, we say [tex]\hat{y}_Y(y^*):=\hat{y}(x^*)[/tex]. Does this depend on the choice of extension [tex]x^*[/tex]?
 

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  • #2
gel
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Given a normed space [tex]X[/tex] over [tex]\mathbb{R}[/tex], if [tex]Y[/tex] is a subspace of [tex]X[/tex], what is the relationship between [tex]Y^*[/tex] and [tex]X^*[/tex]? Can [tex]Y^*[/tex] be identified with some subspace of [tex]X^*[/tex]?
Elements of X* are linear maps X->R, so can also be considered as elements of Y* by restriction to Y.
This gives a canonical map X*->Y*.

The kernel of the map, say K, consists of those elements of X* which map all of Y to 0. Also, the Hahn-Banach theorem says that any element of Y* can be extended to a continuous linear map X->R. So, X*->Y* is onto.

These facts are enough to conclude that Y* is isomorphic to X*/K. i.e. a quotient, not a subspace, of X*.

If X is a Hilbert space, then X* will also be a Hilbert space (isomorphic to X), in which case X* can be decomposed as the sum of K and its orthogonal complement K', which will be isomorphic to Y*. Don't think that X* can be decomposed like this in general though.
 

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