Continuous dual space

Remarks

The notations $E^*$ and $E'$ are both common; some texts reserve $E^*$ for the algebraic dual. A linear form $f$ is continuous if and only if it is bounded on some neighbourhood of $0$. For a normed space $E$, $f$ is continuous if and only if $\norm{f} = \sup_{\norm{x} \le 1} \abs{f(x)} < \infty$, and $E'$ with this norm is a Banach space. Example: on $\ell^2$, for fixed $y \in \ell^2$, the form $f(x) = \sum_n x_n y_n$ is in $(\ell^2)'$, since $\abs{f(x)} \le \norm{y} \norm{x}$ by the Cauchy–Schwarz inequality. Non-example: on the space $c_{00}$ of finitely supported sequences with the sup norm, $f(x) = \sum_n x_n$ is linear but not continuous, since $x = e_1 + \cdots + e_k$ has norm $1$ and $f(x) = k$. For a Hausdorff locally convex space $E$, the Hahn–Banach theorem gives that $E'$ separates the points of $E$; for $L^p([0, 1])$ with $0 < p < 1$, by contrast, $E' = \set{0}$.