For parabolic sobolev spaces I follow the following definition:

According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$

: If we have a function such that $f \in W^{1,2}(I; L^2(\Omega))$ with, in addition, $ \nabla f(x,\cdot) \in L^2(I;L^2(\Omega))$, can we claim that $f \in W^{1,1,2}(I \times \Omega)$?Now my question is

Instinctively I would say yes, but I need a math confirmation.

Any hint or help is much appreciated.

Thanks in advance!