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DualSpace
In linear algebra and functional analysis, given a vector space
A linear functional is a linear map from
- Additivity: For all
$x, y$ in$V$ , we have$f(x + y) = f(x) + f(y)$ - Homogeneity: For all
$x$ in$V$ and all$\alpha$ in$F$ , we have$f(\alpha x) = \alpha f(x)$
If
Now, when
In particular, if
With this norm,
The concept of a dual space is a fundamental one in functional analysis and it leads to many important results, such as the Hahn-Banach theorem, the Riesz representation theorem, and the Banach-Alaoglu theorem.