Density of the canonical embedding $J: E \mapsto E^{**}$

Question

Assume that E is $ \mathbb K$−Banach space and denote by $J : E → E^{∗∗}$ the canonical embedding of $E$ into its bidual $E^{∗∗}$.

By Goldstine Theorem, I know that in this case $J(E)$ is dense in $E^{∗∗}$ with respect to the weak* topology $σ(E^{∗∗},E^∗)$.

But why is $J(E)$ not dense with respect to the norm topology or the weak topology $σ(E^{∗∗},E^{∗∗∗})$? Could someone give me an example of those cases or explain it?

Answer 1

The canonical embedding $J \colon E \to E^{\ast\ast}$ is isometric, i.e. $\lVert J(x)\rVert_{E^{\ast\ast}} = \lVert x\rVert_E$ for all $x\in E$. Since $E$ is a Banach space, it follows that $J(E)$ is a complete subspace of $E^{\ast\ast}$, and therefore closed.

As a linear subspace, $J(E)$ is convex, and a convex subset of a Hausdorff locally convex space is weakly closed if and only if it is closed in the original topology. Thus it follows that $J(E)$ is also $\sigma(E^{\ast\ast}, E^{\ast\ast\ast})$-closed.

Hence $J(E)$ is norm-dense or weakly dense in $E^{\ast\ast}$ only if $E$ is reflexive.

Answer 2

Let $E= c_0(\mathbb{N})$, the set of sequences $(a_n)\in \mathbb{R}^{\mathbb{N}}$ such that $a_n \mapsto 0$ when $n \mapsto \infty$. Define $\|(a_n)\|= \max_{n\in \mathbb{N}} |a_n|$.

Then $E^*=\ell_1(\mathbb{N})$, and $E^{**}= \ell_{\infty}(\mathbb{N})$.

But $E$ is not dense in $E^{**}$ for the norm $\|. \|_{\infty}$