Ok so in a typical "signals and systems" course taught to engineering students (usually electrical engineers, from what I understand), an LTI (linear and time-invariant) system is defined as a "system" (which is just an operator) $T$ that satisfies:
- Linearity: given any two "signals" (real-valued functions of a real variable, usually denoted $x(t)$ where $t$ is interpreted as time) $x_1(t)$ and $x_2(t)$ with system outputs $y_1(t)$ and $y_2(t)$ respectively, and given any two constants $c_1$ and $c_2$ (real or complex-valued), then the system output of $c_1 x_1(t)+c_2 x_2(t)$ is $c_1y_1(t)+c_2y_2(t)$.
- Time invariance: if $y(t)$ is the system output of a signal $x(t)$, then the system output of a shifted version of $x(t)$, i.e. $x(t-t_0)$ as a function of $t$, is $y(t-t_0)$, for any constant $t_0 \in \mathbb{R}$.
From here, it is argued that given an LTI system, the output of $x(t)$ is $y(t) = (h\ast x)(t)$, where $h = T(\delta)$ is the impulse response, i.e. the system response to the Dirac delta function $\delta(t)$ (which as we all know it's not actually a function, but that's not the point here), and $\ast$ denotes convolution, i.e. $(x\ast y)(t) = \int_{-\infty}^{+\infty} x(\tau)y(t-\tau)\,d\tau$.
Now, the justification is usually that we can write $$x(t) = \int_{-\infty}^{+\infty} x(\tau) \delta(t-\tau)\, d\tau$$ and hence, by the linearity assumption, the output $y(t)$ given $x(t)$ must be $$y(t) = \int_{-\infty}^{+\infty} x(\tau) h(t-\tau)\, d\tau.$$
My question now is, for even just a function series $\sum_{k=1}^\infty c_k x_k(t)$, must the output (assuming the series converges) be $\sum_{k=1}^\infty c_k y_k(t)$, with $y_k=T(x_k)$?
After some reflection I arrived at the conclusion, which I was hoping someone could confirm, that the underlying/unstated assumption is that $T$ is a continuous operator (or equivalently, a bounded operator, since it's assumed to be linear), because then we'll have $T(x_n) \to T(x)$ when $n\to\infty$, given $x_n,x$ such that $x_n \to x$. In particular, since a series function is just the limit of its partial sums, then:
\begin{align*} T\left(\sum_{k=1}^\infty c_k x_k\right) &= T\left(\lim_{K\to \infty} \sum_{k=1}^K c_k x_k\right)\\ &= \lim_{K\to \infty} T\left(\sum_{k=1}^K c_k x_k\right)\\ &= \lim_{K \to \infty} \sum_{k=1}^K c_k T(x_k)\\ &= \sum_{k=1}^\infty c_k y_k \end{align*}
And then the integral could be interpreted as a series, at least for the Riemann-integrable functions I suppose. Why then isn't the continuity/boundedness assumption on the operator stated anywhere? Is it because non-continuous linear operators are somewhat esoteric? I'm fairly new to functional analysis so maybe I'm wrong in this last part.