For a linear system Au = b, if A is a square matrix in Hilbert space we can write $A \in \mathbb{R}^{H\times H}$.
However if u is time-dependent, can we write something like $u \in \mathbb{R}^{H\times T}$ where T denotes time?
I'm new to Math Stack Exchange, please forgive me if it's too simple, many thanks!
You can write $u \colon \mathbb{R} \to H$, since (if I understand you correctly) $u(t)$ is an $H$-valued function of $t \in\mathbb{R}$.