# The Canonical Embedding of X into X**

For each $x \in X$ define a function $J_x : X^* \to \mathbb{C}$ for all $\varphi \in X^*$ by:

(1)That is, $J_x$ takes each continuous linear functional $\varphi$ in $X^*$ and evaluates it at $x$. It is easy to see that for each $x \in X$, $J_x$ is a linear functional on $X^*$. To show this, let $\varphi, \psi \in X^*$ and let $\lambda \in \mathbb{C}$. Then:

(2)So indeed, $J_x : X^* \to \mathbb{C}$ is a linear functional on $X^*$, so each $J_x \in X^{*\#}$. Furthermore, each $J_x$ is continuous. We have that:

(4)(Where the inequality at $(\dagger)$ comes from $\varphi : X \to \mathbb{C}$ being a continuous linear functional on $X$ and $\| x \|$ is a fixed number).

Hence, for each $x \in X$, $J_x \in X^{**}$.

We now define another function. Let $J : X \to X^{**}$ be defined for each $x \in X$ by:

(5)Then $J$ is a linear function since for all $x, y \in X$ and for all $\lambda \in \mathbb{C}$ we have that:

(6)Definition: The linear operator $J : X \to X^{**}$ defined above is the Canonical Embedding of $X$ into $X^{**}$. |

Later we will see that $J$ is in fact an isometry and hence is continuous and injective.