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.