Normed space and its double dual

[dmuthuk]

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 X over \mathbb{R}, if Y is a subspace of X, what is the relationship between Y^* and X^*? Can Y^* be identified with some subspace of X^*?

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

 

[gel]

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

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.