double dual

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Double Dual

The double dual of a vector space $V$ is $V^{″} = (V^{'})^{'}$ .

Example

Let $v \in V$ . Define $T_{v} : V^{'} \to C$ by

T_{v} (v^{'}) = v^{'} v, v^{'} \in V^{'}

Then $T_{V} \in V^{″}$ .

Clearly $T_{v}$ is linear
$T_{v}$ is bounded since $| T_{v} (v^{'}) | = | v^{'} (v) | \leq | | v^{'} | | \times \underset{constant}{\underset{⏟}{| | v | |}}$

$⟹ T_{v} \in (V^{'})^{'} = V^{″}$ and $| | T_{v} | | \leq | | v | |$

File	Last Modified
Functional Analysis Lecture 6	2025-07-08
the functional to the double dual is isometric	2025-06-12

double dual

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