When I have a PDE $$ u_t+\Delta u=f, \ \ x\in\Omega $$ with $H:=H^2(\Omega) $
$f:[0,T]\to H $
what does it mean that $f\in L^2(0,T;H)$ for every $T>0$? (what is dot comma?)
what does it mean that we are looking for $u\in C([0,T];L^2(\Omega))$?
thank you
The typographic mark ";" is called a semicolon and it functions as a logical separator here. It is stronger than a comma. It is common in lists where it is used to separate comma-separated values. The notation $C(X, Y)$ is standard for "continuous maps from $X$ to $Y$", so $C(X; Y)$ probably means the same thing. As for $L^2(0, T; H)$, it probably means "square-integrable maps from the interval $[0, T]$ to the space $H$".