I am working through some fluid mechanics and I can't seem to find a precise definition anywhere for this function space notation:
$$z(\alpha,t) \in C^1 \left( [-T,T]; C[0,2\pi] \right)$$
I am specifically looking at the bottom of page 332 (Proposition 8.6) in Majda and Bertozzi's Vorticity and Incompressible Flow. I have seen several other examples of this notation such as
$$L^\infty (0,T;C^{1,\alpha})$$ and $$C(0,T;H^1).$$
I think I get the general idea - for instance $L^\infty (0,T;C^{1,\alpha})$ is something like the set of functions $f(t,x)$ that are $L^\infty$ in time on the interval $[0,T]$ and $C^{1,\alpha}$ in space (could be wrong about that), but it also seems that the Holder norm is interacting with the $L^\infty$ norm somehow.
Does anybody have a good definition of these kind of spaces or know of a reference that defines them thoroughly?
It means that the function $z(\alpha, t)$ is regarded as a $C^1$ function of $\alpha$ (in $C^1[0, 2\pi])$) for each $t \in [-T,T]$: $t \mapsto z_{\alpha}(t):=z(\alpha,t)$. You would read it as "$C^1$ function in $[-T,T]$, with values in $C^1[0, 2\pi]$".