# Examples of interesting cones in infinite-dimensional Hilbert spaces

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.

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.

• Thanks! Can you add some details on whether X is a Hilbert space here? A reference would be great. Jun 14, 2021 at 7:27
• Yes $X$ is a Hilbert space, I have added this information and I also added that $X$ is called a Sobolev space. As reference, you can take any book on Sobolev spaces.
– gerw
Jun 14, 2021 at 9:11
• Any favourite reference easily accessible to outsiders, gerw? Also, you say that $X^*=H^{-1}(\Omega)$. But $X^*=X$ for a Hilbert space - what am I missing? Jun 14, 2021 at 16:09
• Maybe "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Brezis; in German there is also "Lineare Funktionalanalysis" by Alt. In $X^* = H^{-1}(\Omega)$ (and similar statements), the "=" is actually a "is isometrically isomorphic"; and all separable Hilbert spaces are isometrically isomorphic. For function spaces, one typically does not use the isomorphism which makes $X = X^*$, since this involves the inner product and this depends on the space. For example, we still identify $L^2(\Omega)^* = L^2(\Omega)$, but not $H^1(\Omega)$ and its dual.
– gerw
Jun 15, 2021 at 5:51