In the definition, the unbounded signal (or data) is the signal that can take infinite value and to make it bounded we can normalize it and then it will $\in L_\infty$.
In control theory, and especially in adapive control if we have the following system: $y(t)=\theta^*u(t)$
where $\theta^*$ is unknown parameter; $u$ and $y$ are measurment signals but they are not necessary to be bounded
To estimate the unknown parameter $\theta^*$ we present estimate model:
$\hat{y}(t)=\theta\hat{u}(t)$
where $\theta$ is the estimating of $\theta^*$
The update law or adaptive law that can give the estimation we can generate it by using the following minimization problem of gardient method:
$J=\frac{1}{2}(y-\theta u)^2$
where we have to minimize the cost function with respect to $\theta$ But If the signals $u$ and $y$ not guranteed to be bounded then the minimization problem is ill-posed, therefore in the estimating process we have to normalize the signals $u$ and $y$ to make it bounded and that as following:
$\bar{y}=\frac{y}{m}$
$\bar{u}=\frac{u}{m}$
where $m$ is the normalizing signal it is given by : $m^2=1+n^2$ and we choose $n$ such that $\frac{y}{m},\frac{u}{m}\in L_\infty$ i.e. to be bounded.
the question: if $u$ and $y$ unbounded, why the minimization problem is ill-posed?
You may consider $u$ and $y$ as a gauge, and if they are not bounded then simply your function will change freely. If you want to optimize, $u$ and $y$ can vary and result in failure to optimize.