Normally, when we talk about Gelfand triple we have three Hilbert spaces $$\newcommand{\X}{\mathcal{X}} \X_+ \subset \X_0 \subset \X_- $$ such that the subsets are dense, the embedding mappings are continuous, and $\X_-$ is the dual space of $\X_+$ with respect to the inner product of $\X_0$.
I want to weaken some conditions. I want to drop the chain of inclusion and the continuity of the embeddings. Instead I have that the intersection $\X_+ \cap \X_0$ is dense in $\X_0$ with respect to $\Vert . \Vert_{\X_0}$ and dense in $\X_+$ with respect to $\Vert . \Vert_{\X_+}$, and respectively for $\X_-$. This looks like
where $D_+ := \X_+ \cap \X_0$ and $D_- := \X_- \cap \X_0$. Still, $\X_-$ is the dual space of $\X_+$ with respect to the inner product of $\X_0$. This means that for $f \in D_+$ and $g \in D_-$ we have $$ \langle f,g \rangle_{\X_+,\X_-} = (f,g)_{\X_0}. $$
I would call it generalized Gelfand triple. Is there any theory about this scenario?
