Is $f \in C^1([0,T]\times \Omega) \iff f \in C([0,T]; C^1(\Omega))$
My guess is no, I think the RHS is a bit stronger. But I can't show it. Can someone help me please?
Is $f \in C^1([0,T]\times \Omega) \iff f \in C([0,T]; C^1(\Omega))$
My guess is no, I think the RHS is a bit stronger. But I can't show it. Can someone help me please?
If $f \in C^1([0,T]\times \Omega)$, then $\frac{\partial}{\partial t} f \in C^([0,T] \times \Omega)$. Functions in the set on the right do not have this property.
On the other hand, if $\Omega$ is compact, then $\nabla f C^([0,T] \times \Omega$ is uniformly continuous, hence $\nabla f \in C([0,T],C(\Omega))$ and therefore $f \in C([0,T],C^1(\Omega))$.
So the left set is usually smaller.