To review for an exam, I'm trying to write up a short proof of the following:
Let $J: X \rightarrow X^{**}$ be the natural embedding of the normed
linear space $X$ into its bidual $X^{**}$, given by $J(x) = f(x)$. This embedding is a linear and isometric.
The Hahn-Banach theorem gives us $\phi \in X^*$ (a linear functional on $X$) which $\| \phi \| =1$ and $f(x) = \|x\|$. This implies $||x|| \leq \|J(x)\|$.
I have difficulty in following the proofs demonstrating that the embedding is bounded, that $\|J(x)\| \leq \|x\|.$ How can this be proven in a short manner without being "handwaving in manner"?
Best Answer
Let $ \mathbb{S}(X^{*}) $ denote the unit sphere in $ X^{*} $. Then \begin{align} \| J(x) \|_{X^{**}} &\stackrel{\text{def}}{=} \sup_{\varphi \in \mathbb{S}(X^{*})} |[J(x)](\varphi)| \\ &= \sup_{\varphi \in \mathbb{S}(X^{*})} |\varphi(x)| \\ &\leq \sup_{\varphi \in \mathbb{S}(X^{*})} \| \varphi \|_{X^{*}} \cdot \| x \|_{X} \\ &= \| x \|_{X}. \quad (\text{As $ \| \varphi \|_{X^{*}} = 1 $ for all $ \varphi \in \mathbb{S}(X^{*}) $.}) \end{align}