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