# dual space separates points

The following result is a corollary of the Hahn-Banach theorem^{}.

Theorem - Let $X$ be a normed vector space^{}. Given a linearly independent^{} set $\{{x}_{1},\mathrm{\dots},{x}_{n}\}\subset X$ there exist continuous^{} linear functionals^{} ${f}_{1},\mathrm{\dots},{f}_{n}\in {X}^{\prime}$ such that

$${f}_{j}({x}_{k})={\delta}_{jk}\mathit{\hspace{1em}},1\le j,k\le n$$ |

If $x\in span\{{x}_{1},\mathrm{\dots},{x}_{n}\}$, then $x={\displaystyle \sum _{j=1}^{n}}{f}_{j}(x){x}_{j}$.

The above theorem shows that if $f(x)=f(y)$ for every continuous linear functional $f$ then $x=y$, therefore the dual space^{} ${X}^{\prime}$ separates the points of $X$.

Title | dual space separates points |
---|---|

Canonical name | DualSpaceSeparatesPoints |

Date of creation | 2013-03-22 17:30:55 |

Last modified on | 2013-03-22 17:30:55 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 6 |

Author | asteroid (17536) |

Entry type | Corollary |

Classification | msc 15A99 |