Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the distributional sense) is zero.

Is there any workable description of $N^{**}$?

Of course, even the description of $L_1^{**}$ is elusive, yet somewhat concrete (finitely additive measures on the spectrum of $L_\infty$).

• The annihilator of $N$ is called the space of closed currents.
• $N$ is not complemented in $L_1(\mathbb R^n, \mathbb R^n)$ and $N^\perp$ is not complemented in $L_\infty(\mathbb R^n, \mathbb R^n)$.
• $N$ is not an $L$-summand in $N^{**}$ (Godefroy--Lerner).

Is $N$ complemented in $N^{**}$?

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the distributional sense) is zero.

Is there any workable description of $N^{**}$?

Of course, even the description of $L_1^{**}$ is elusive, yet somewhat concrete (measures on the spectrum of $L_\infty$).

• The annihilator of $N$ is called the space of closed currents.
• $N$ is not complemented in $L_1(\mathbb R^n, \mathbb R^n)$ and $N^\perp$ is not complemented in $L_\infty(\mathbb R^n, \mathbb R^n)$.
• $N$ is not an $L$-summand in $N^{**}$ (Godefroy--Lerner).

Is $N$ complemented in $N^{**}$?