The Canonical Embedding J is an Isometry

The Canonical Embedding J is an Isometry

Recall from the Isometries on Normed Linear Spaces page that if $X$ and $Y$ are normed linear spaces then a linear map $T : X \to Y$ is said to be an isometry if:

(1)
\begin{align} \quad \| T(x) \| = \| x \|, \quad \forall x \in X \end{align}

Recall that the canonical embedding of a normed linear space $X$ into the topological double dual $X^**$ is the function $J : X \to X^{**}$ defined for all $x \in X$ by:

(2)
\begin{align} \quad J(x) = J_x \end{align}

Where $J_x : X^* \to \mathbb{C}$ is defined for all $\varphi \in X^*$ by:

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

Then following theorem tells us that the canonical embedding $J$ is in fact as isometry.

Theorem 1: Let $X$ be a normed linear space. Then the canonical embedding $J : X \to X^{**}$ is an isometry.
  • Proof: Let $x \in X$ and consider the linear functional $J_x : X^* \to \mathbb{C}$. Then for all $\varphi \in X^*$ we have that:
(4)
\begin{align} \quad \| J(x)(\varphi) \| = \| J_x(\varphi) \| = \| \varphi(x) \| \leq \| \varphi \| \| x \| = \| x \| \| \varphi \| \end{align}
  • Let $M = \| x \|$ (observe that $\| x \|$ is fixed here). Then:
(5)
\begin{align} \quad \| J(x)(\varphi) \| \leq M \| \varphi \| \end{align}
  • By choosing $\varphi \in X^*$ such that $\| \varphi \| = 1$ we see that:
(6)
\begin{align} \quad \| J(x) \| \leq \| x \| \quad (*) \end{align}
  • Now for each $x \in X$ there is a $\varphi \in X^*$ such that $\varphi (x) = \| x \|$ and $\| \varphi \| = 1$. But $\varphi(x) = J(x)(\varphi)$. Hence:
(7)
\begin{align} \quad \| x \| \leq \| J(x) \| \quad (**) \end{align}
  • From $(*)$ and $(**)$ we conclude that for each $x \in X$ we have that:
(8)
\begin{align} \quad \| J(x) \| = \| x \| \end{align}
  • So $J$ is an isometry.
Corollary 2: Let $X$ be a normed linear space. Then the canonical embedding $J$ is continuous and injective.
  • Proof: This follows immediately by the theorems on the page referenced above. $\blacksquare$
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