Questions About Continuous-Variable Quantum Computing

In linear we talk about linear maps $T \in Hom (V, W)$ for vector spaces $V, W$ , but is there any idea of a span of linear maps? Namely, given a set $β = {T_{1}, . . ., T_{n}}$ is there a way to talk about the set of possible transformations $β^{'} = {T_{i_{1}}^{r_{1}} T_{i_{2}}^{r_{2}} \dots T_{i_{n}}^{r_{n}} : \forall 1 \leq k \leq n (1 \leq i_{k} \leq n, r_{k} \in Q)}$ in the language of linear algebra (or even functional analysis), where we are allowed to apply any $T_{i}$ any rational amount of times (I want to allow things like $\sqrt{T_{1}}$ and such, I know I know we have to assume that $T_{1}$ is positive but I then we can turn each $T_{1}$ into $S_{1} \sqrt{T^{*} T}$ where $S_{1}$ is an isometry, so then we could technically extend this idea )? This idea comes up a lot in the context of any type of computing (quantum or classical). I've tried to do research on this idea but I've found nothing on it yet. For this answer let's just consider the case where $F = C$ for simplicity.
What's the most apparent properties do we gain/lose once a vector space $V$ becomes (countably or uncountably infinitely) dimension, ie: $∄ i, \dim (V) = i$ .
How does the Spectral Theorem (both real and complex) update when talking in functional analysis terms, so when the dimension becomes infinite?