Definition:Evaluation Linear Transformation/Normed Vector Space
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Definition
Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\Bbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual of $\struct {X, \norm \cdot_X}$.
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm \cdot_X}$.
For each $x \in X$, define $x^\wedge : X^\ast \to \Bbb F$ by:
- $\map {x^\wedge} f = \map f x$
Then we define the evaluation linear transformation from $X$ into $X^{\ast \ast}$ as the function $\iota : X \to X^{\ast \ast}$ defined by:
- $\map \iota x = x^\wedge$
for each $x \in X$.
Also see
- Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual shows $\iota$ is indeed a function $X \to X^{\ast \ast}$
- Evaluation Linear Transformation on Normed Vector Space is Linear Isometry shows that $\iota$ can be used to identify $X$ with a subset of $X^{\ast \ast}$.
- Results about evaluation linear transformations can be found here.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $26.1$: The Second Dual