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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]?

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]?