Definition:Evaluation Linear Transformation/Normed Vector Space

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 $X^{\ast \ast}$.