Examples of interesting cones in infinite-dimensional Hilbert spaces

Question

Let $X$ be a real Hilbert space, and let $K$ be a closed convex cone.

The dual cone is defined by $K^*=\{x^*\in X \mid (\forall k\in K)\, \langle x^*,k\rangle \geq 0 \}$.

I am looking for some interesting examples of $K$ and $K^*$, especially when $X$ is infinite-dimensional.

For example,

• $X=\ell_2$ and $K=\ell_2^+ = K^*$
• $X=L_2[0,1]$ and $K=L_2^+[0,1]=K^*$

Note that these examples satisfy $K-K=X$. I wonder if there are some concrete examples where

$$K-K\neq X\quad \text{but}\quad \overline{K-K}=X$$

and $K^*$ is actually known?

Any examples/comments/references would be greatly appreciated.

Answer 1

Let us take the Sobolev space $X = H_0^1(\Omega)$ and $K = X^+ = \{ v \in H_0^1(\Omega) \mid v \ge 0 \text{ a.e.}\}$. Then, we have $X = K - K$. Note that $X$ is a Hilbert space.

However, the dual cone $K^*$ coincides with the non-negative functionals in $H^{-1}(\Omega) = X^*$. This are precisely the functionals in $X^*$ which can be represented by measures which are finite on compact sets, i.e., for each $\mu \in K^*$, there is a measure $\hat\mu$ such that $$ \langle \mu, v\rangle = \int_\Omega v \, \mathrm{d}\hat\mu\qquad \forall \mu \in H_0^1(\Omega) \cap C_c(\Omega).$$

Now, one can check that we have $X^* = \overline{K^* - K^*} \ne K^* - K^*$, see also Decomposition of functionals on sobolev spaces and Decomposition of measures acting on sobolev spaces.