If $Q$ is an operator on a Hilbert space $U$, is there a name for the space $Q^{1/2}U$ equipped with $(u,v)↦\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U$?

Question

Let $U$ be a $\mathbb R$-Hilbert space and $Q$ be a bounded, linear, nonnegative and self-adjoint operator on $U$.

There is a "canonical" way to define an inner product on $U_0:=Q^{1/2}U$, namely, via $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }u,v\in U_0$$ (where $Q^{-1/2}$ denotes the pseudoinverse of $Q^{1/2}$, if $Q$ is not injective). Is there a name for this space?