Definition:Evaluation Linear Transformation
Definition
Module Theory
Let $R$ be a commutative ring with unity.
Let $G$ be an $R$-module.
Let $G^*$ be the algebraic dual of $G$.
Let $G^{**}$ be the double dual of $G^*$.
For each $x \in G$, we define the mapping $x^\wedge: G^* \to R$ as:
- $\forall t \in G^*: \map {x^\wedge} t = \map t x$
The mapping $J: G \to G^{**}$ defined as:
- $\forall x \in G: \map J x = x^\wedge$
is called the evaluation linear transformation from $G$ into $G^{**}$.
It is usual to denote the mapping $t: G^* \to R$ as follows:
- $\forall x \in G, t \in G^*: \innerprod x t := \map t x$
Normed Vector Space
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 $x \in X$.