Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?

Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.

If we define a collection of vector fields $\Gamma$ from $G^{0}$ to $H=\cup_{x\in X}H_{x}$ such that for $f\in \Gamma$, $f(x)\in \mathbb{B}_{x}$, then obviously $x\to \|f(x)\|_{H_{x}}$ is a continuous function since $\|f(x)\|_{H_{x}}=1$ for every $x$.

So Can this family $\{H_{x}\}$ form a continuous field of Hilbert space with the continuous sections being the linear span of elements in $\Gamma$?

Here, is the definition of a continuous field of Banach space: