Topological Dual Space

linear-algebra

Definition

Topological Dual Space

Let $(V,\Vert \cdot \Vert )$ be a normed vector space over a field $\mathbb{F}$ (typically $\mathbb{R}$ or $\mathbb{C}$). The topological dual space $V^{\ast }$ is the vector space of all continuous linear functionals $f:V\to \mathbb{F}$:

$$V^{\ast }=\{f:V\to \mathbb{F}\mid f\text{ is linear and continuous}\}$$

Equipped with the operator norm:

$$\Vert f{\Vert }_{V^{\ast }}=\underset{\Vert v\Vert \le 1}{\sup }|f(v)|=\underset{v\ne 0}{\sup }\frac{|f(v)|}{\Vert v\Vert }$$

Algebraic vs Topological Dual

The algebraic dual $V^{\prime }$ contains all linear functionals, continuous or not. For infinite-dimensional spaces:

$$V^{\ast }⊊V^{\prime }$$

Continuity is required because discontinuous linear functionals exist only in infinite dimensions (using Hamel bases) and are pathological for analysis.