The Canonical Embedding of X into X**

# 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)
\begin{align} \quad (J_x)(\varphi) = \varphi (x) \end{align}

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)
\begin{align} \quad (J_x)(\varphi + \psi) = (\varphi + \psi)(x) = \varphi(x) + \psi (x) = J_x(\varphi) + J_x(\psi) \end{align}
(3)
\begin{align} \quad (J_x)(\lambda \varphi) = (\lambda \varphi)(x) = \lambda \varphi (x) = \lambda J_x (\varphi) \end{align}

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)
\begin{align} \quad \| J_x(\varphi) \| = \| \varphi (x) \| \overset{\dagger} \leq \| \varphi \| \| x \| = \| x \| \| \varphi \| \end{align}

(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)
\begin{align} \quad J(x) = J_x \end{align}

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

(6)
\begin{align} \quad J(x + y) = J_{x + y} = J_x + J_y = J(x) + J(y) \end{align}
(7)
\begin{align} \quad J(\lambda x) = J_{\lambda x} = \lambda J_x = \lambda J(x) \end{align}
 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.