Meaning of uniform convergence for a one parameter family of functions

Question

Given a certain function $y(t)$, consider the functions $y_\varepsilon(t)$ as $\varepsilon$ varies in $\mathbb{R}$. It is not relevant to know the expression of these functions. I'm asked to prove that "$y_\varepsilon(t)$ uniformly converges to $y(t)$ on every interval $[-T,T]$". Since it's not a sequence of functions, what's the meaning of uniform convergence in this case? I think I have to prove that $$\lim_{\varepsilon \to 0}\|y_\varepsilon-y\|_\infty =0$$ where $$\|y_\varepsilon-y\|_\infty =\sup_{t\in[-T,T]}\|y_\varepsilon(t)-y(t)\| $$ Is this correct?

Answer 1

Yes. Since you are dealing with a normed space it is sufficient to show that $\lim_n ||y_{\varepsilon_n}-y||_\infty = 0$ for every seqence $\varepsilon_n$ converging to $0$. Only the notation you have been using is not common (and slightly incorrect), the norm applies to functions, not to functions evaluated at some $t$ (remove the $t$).