I am now learning the following version of the resonance theorem.
Let $X, Y$ be Banach spaces. Let $\{T_\lambda: \lambda\in\Lambda\}$ be a family of bounded linear operators from $X$ to $Y$. If $\{T_\lambda: \lambda\in\Lambda\}$ is pointwise bounded on $X$, then $\{T_\lambda: \lambda\in\Lambda\}$ is uniformly bounded, i.e., $\sup\|T_\lambda\|<\infty$. Or equivalently, if $\sup\|T_\lambda\|=\infty$, then there exist $x\in B$ such that $\sup\|T_\lambda x\|=\infty$.
For those who don't think it can be called "resonance", see theorem 1.16 here: My statement is equivalent to this.
Also, in physics, we know that resonance occurs when the imposed frequency onto an oscillating system equals its natural frequency.
Question: How is the resonance theorem related to resonance in physics?