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 $x \in X$.