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.(adsbygoogle = window.adsbygoogle || []).push({});

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]?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Normed space and its double dual

Loading...

Similar Threads - Normed space double | Date |
---|---|

A For finite dimension vector spaces, all norms are equivalent | Sep 29, 2016 |

Normed linear space vs inner product space and more | Oct 15, 2013 |

Variation of the triangle inequality on arbitrary normed spaces | Oct 12, 2013 |

Norms of a general vector space | Aug 24, 2011 |

Hilbert space and infinite norm vectors | Dec 17, 2004 |

**Physics Forums - The Fusion of Science and Community**