# Generalized Gelfand Triples

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 $$\langle f,g \rangle_{\X_+,\X_-} = (f,g)_{\X_0}.$$