Question:
What is a formal definition of a discrete LTI system?
In many textbooks on signal and system theory no rigorous definition is stated at all. One (naive) idea is to define a DLTI as any linear operator $T:\mathbb{C}^{\mathbb{Z}} \to \mathbb{C}^{\mathbb{Z}}$ that is shift-invariant, i.e. $$TSx = STx \quad \text{for all $x \in \mathbb{C}^{\mathbb{Z}}$}$$ where $S:\mathbb{C}^{\mathbb{Z}} \to \mathbb{C}^{\mathbb{Z}}$ is the shift operator, i.e. $(Sx)^{(n)} = x^{(n-1)}$. However, we also want that the operator $T$ is completely determined by its impulse response, $h = T \delta_{0}$ where $\delta_k^{(n)} = 1$ for $n = k$ and $0$ otherwise. In other words, we want $$x = \sum_{k \in \mathbb{Z}} x^{(k)} \delta_k \quad \Rightarrow \quad Tx = \sum_{k \in \mathbb{Z}} x^{(k)} T \delta_k = \sum_{k \in \mathbb{Z}} x^{(k)} S^k h$$ to hold, from which we can infer that $T$ should be continous in the topology of pointwise convergence on $\mathbb{C}^{\mathbb{Z}}$. Is this the "right" definition of a discrete LTI system?