# Definition:Normed Dual Space

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## Definition

Let $\struct {X, \norm \cdot_X}$ be a normed vector space.

Let $X^\ast$ be the vector space of bounded linear functionals on $X$.

Let $\norm \cdot_{X^\ast}$ be the norm on bounded linear functionals.

We say that $\struct {X^\ast, \norm \cdot_{X^\ast} }$ is the **normed dual space** of $X$.

## Also known as

The **normed dual space** of $X$ may be known as the **normed dual**, **continuous dual** (in view of Continuity of Linear Functionals) or simply **dual** of $X$.

## Also see

- Definition:Topological Dual Space is an extension of this concept general toplogical vector spaces.

- Results about
**normed dual spaces**can be found**here**.

## Linguistic Note

The **normed dual space** is not to be confused with the algebraic dual space of $X$ (which may also be referred to as the dual of $X$) which is the space of all linear functionals on $X$, not just those that are bounded.

## Sources

- 2020: James C. Robinson:
*Introduction to Functional Analysis*... (previous) ... (next) $12.1$: The Dual Space