# dual space separates points

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

Let $X$ be a normed vector space. Given a linearly independent set $\{x_{1},\dots,x_{n}\}\subset X$ there exist continuous linear functionals $f_{1},\dots,f_{n}\in X^{\prime}$ such that

 $f_{j}(x_{k})=\delta_{jk}\;\;\;\;,1\leq j,k\leq n$

If $x\in span\{x_{1},\dots,x_{n}\}$, then $\displaystyle x=\sum_{j=1}^{n}f_{j}(x)x_{j}$.

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 DualSpaceSeparatesPoints 2013-03-22 17:30:55 2013-03-22 17:30:55 asteroid (17536) asteroid (17536) 6 asteroid (17536) Corollary msc 15A99