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dual space separates points

(Corollary)

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

Theorem - Let be a normed vector space. Given a linearly independent set $\{x_1,\dots ,x_n\} \subset X$ there exist continuous linear functionals $f_1, \dots , f_n \in X'$ such that

$\displaystyle f_j(x_k)=\delta_{jk}\;\;\;\; , 1\leq j, k \leq n$

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 for every continuous linear functional then , therefore the dual space separates the points of .

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Cross-references: points, dual space, linear functionals, continuous, linearly independent, normed vector space, Hahn-Banach theorem
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This is version 3 of dual space separates points, born on 2007-08-29, modified 2007-08-31.
Object id is 9906, canonical name is DualSpaceSeparatesPoints.
Accessed 567 times total.

Classification:

15A99 (Linear and multilinear algebra; matrix theory :: Miscellaneous topics)

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