$\DeclareMathOperator\Lip{Lip}\DeclareMathOperator\AE{AE}$**Background**


*Gelfand triples.* Let $\mathcal B$ be a Banach space, $\mathcal B^\*$ its dual space, and $\mathcal H$ a Hilbert space. The triple $(\mathcal B,\mathcal H, \mathcal B^\*)$ is called a Gelfand triple if the following embeddings are continuous
$$
\mathcal B \hookrightarrow \mathcal H \hookrightarrow \mathcal B^\*.
$$


An example that I am familiar with is the triple $(BV(\Omega), L^2(\Omega), BV^\*(\Omega))$, where $\Omega \subset \mathbb R^2$ is bounded and $BV(\Omega)$ is the space of functions of bounded variation.


*Arens-Eells space.* Let $X$ be a compact pointed metric space with base point $e$. An *elementary molecule* is defined as follows (Nik Weaver, Lipschitz Algebras, 2nd ed.)
$$
m\_{pq} := \delta\_p - \delta\_q,
$$
where $\delta\_p, \delta\_q$ are delta-functions placed at $p,q$.


The Arens-Eells space $\AE(X)$ (also known as the Lipschitz-free space) is the completion of the linear span of elementary molecules with respect to the Arens-Eells norm
$$
\|{m}\|\_{\AE} := \inf \left\{\sum\_{i=1}^n |{a\_i}| d(p\_i,q\_i) \colon m = \sum\_{i=1}^n a\_i m\_{p\_iq\_i} \right\},
$$
where $d(p,q)$ is the distance between $p,q \in X$.


The dual of the Arens-Eells space is the $\Lip\_0(X)$ space of all Lipschitz functions on $X$ vanishing at $e$, equipped with the following norm
$$
\|f\|\_{\Lip\_0} := \Lip(f),
$$
where $\Lip(f)$ denotes the Lipschitz constant.


**Question**


Is there a Hilbert space $\mathcal H$ such that $(\AE(X), \mathcal H, \Lip\_0(X))$ form a Gelfand triple? It would suffice for me to think of $X$ as the unit ball in $\mathbb R^n$ with base point $0$, equipped with the euclidean metric.



Existence of a Gelfand triple involving the Arens–Eells space (aka Lipschitz free space)
Your question is equivalent to asking if there is an injective bounded linear operator from $AE(X)$ into a Hilbert space when $X$ is a compact metric space. The answer is "yes" because $AE(X)$ is separable. It is elementary to construct a nuclear injective linear operator from an arbitrary separable Banach space into a Hilbert space.


