dual space separates points

The following result is a corollary of the Hahn-Banach theoremMathworldPlanetmath.

Theorem - Let X be a normed vector spacePlanetmathPlanetmath. Given a linearly independentMathworldPlanetmath set {x1,,xn}X there exist continuousPlanetmathPlanetmath linear functionalsMathworldPlanetmathPlanetmath f1,,fnX such that


If xspan{x1,,xn}, then x=j=1nfj(x)xj.

The above theorem shows that if f(x)=f(y) for every continuous linear functional f then x=y, therefore the dual spaceMathworldPlanetmathPlanetmathPlanetmath X 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